$\mathrm{PGL}(3)$-invariant integrable systems from factorisation of linear differential and difference operators

PGL(3)-in v arian t in tegrable systems from factorisation of linear diﬀeren tial and diﬀerence op erators F rank Nijhoﬀ , Lin yu Peng , Cheng Zhang , Da-jun Zhang Abstract In this pap er, we present a uniﬁed approac h to constructing contin uous and discrete PGL(3)-in v ariant in tegrable systems, formulated in terms of the common dep enden t v ariables z 1 , z 2 , from linear sp ectral problems and their factorisation. Starting from third-order spectral problems, w e ﬁrst pro vide explicit forms of the diﬀerential and dif- ference inv ariants, generalising the Sch warzian deriv ative and cross-ratio to the rank-3 setting. The factorisation induces dualities among linear sp ectral problems, underlying the exact discretisation and multi-dimensional consistency of the associated Boussinesq systems. Then, w e derive b oth contin uous and discrete PGL(3)-in v ariant Boussinesq systems, representing natural rank-3 generalisations of the Sch w arzian KdV and cross- ratio equations. A geometric lifting-decoupling mec hanism is developed to explain the reduction of these systems to the PGL(2)-inv arian t Boussinesq equations. Finally , w e deriv e a PGL(3)-inv ariant system of gener ating PDEs together with its Lagrangian structure, in which the lattice parameters serve as indep enden t v ariables, providing the generating PDE system for the Boussinesq hierarc hy . 1 In tro duction Originating in the geometric in vestigations of Klein, Lie, and Hilbert, the theory of diﬀer- ential invariants sough t to characterise geometric ob jects preserv ed under group actions. In the con text of integrable systems, suc h in v arian ts naturally emerge from linear sp ectral problems asso ciated with Lax pairs, where the underlying symmetry groups act on the spaces of solutions. Remark ably , the nonlinear evolution equations that arise as “pro jec- tiv e formulations” of the Lax pairs typically inherit this inv ariance, leading to so-called “Sc hw arzian” or “pro jective” v ersions of w ell-known in tegrable mo dels. The paradigmatic example is the Schwarzian derivative arising from the second-order Sc hr¨ odinger sp ectral problem. This fundamental PGL(2)-inv ariant appears in the “pro- jectiv e” formulation of the celebrated Korteweg–de V ries (KdV) equation, kno wn as the Schwarzian KdV equation [28, 66]. The c omp osition rule of the Sch w arzian deriv ativ e of- fers a pro jective geometric p erspective on the KdV hierarc hy , reﬂecting the link b et ween pr oje ctive c onne ctions and Vir asor o algebr a with the latter coinciding with the second Hamiltonian structure of the KdV hierarc hy (see, for example, [10, 26, 50, 57]). In the dis- crete realm, the cr oss-r atio e quation [41]—which reapp eared as the Q1( δ = 0) member of the Adler–Bobenko–Suris (ABS) classiﬁcation [2]—serv es as the discrete Sch warzian KdV equation [39]. It plays a central role in the theories of discrete surfaces and discrete holo- morphic functions [3, 5, 7], as well as in reductions to discrete P ainlev´ e equations [40, 45]. More than a mere geometric reform ulation, the Sch warzian KdV equation ma y b e seen as the most fundamental member of a Miura-related chain of KdV equations follo wing Wilson, who also ga ve the name “Ur-KdV” [69] (see [9, 39] for other “Ur-t yp e” equations). Wilson explicitly explained, from a diﬀerential Galois theory p ersp ectiv e, that the KdV 1 p oten tial arises as the PGL(2)-inv ariant subﬁeld of a diﬀerential ﬁeld generated b y the pro jective v ariable in terms of the Sc hw arzian deriv ative, and that the Miura chain corre- sp onds to a tow er of diﬀeren tial ﬁeld extensions where the pro jectiv e v ariable sits at the top. Iden tifying PGL(2) as the diﬀeren tial Galois group, a natural Poisson structure on the space of pro jectiv e curves w as constructed using the theory of Poisson–Lie groups [36]. Remark ably , this framew ork was extended to the discrete setting, with explicit construc- tions of diﬀerence ﬁeld extensions asso ciated with discrete sp ectral problems and P oisson structures on the space of discrete pro jectiv e curves (p olygons) [35, 36] (see also [25, 34]). Despite these adv ances in the rank-2 KdV setting, a systematic pro jective formulation for the rank-3 Boussinesq (BSQ) equations remains surprisingly underdev elop ed. Within the Gel’fand–Dikii formalism, the BSQ hierarc hy is w ell understo od [10, 12, 42], and even the PGL(2)-inv ariant forms of BSQ, kno wn as Schwarzian BSQ e quations , ha v e long b een kno wn in b oth contin uous and discrete cases [39, 66]. Ho w ever, a pro jectiv e form ulation of BSQ equations in terms of PGL(3)-inv ariants, where PGL(3) arises as the natural group of transformations in the rank-3 setting, has not b een fully established. The relev ance of PGL(3) pro jectiv e formulation of BSQ is further underscored by recen t dev elopments in discrete integrable systems. On the one hand, the construction of p entagr am maps is ro oted in the geometry of discrete pro jective curves in the pro jectiv e space P 2 [49, 56]. Indeed, the BSQ equations and p en tagram maps are closely related: con tinuous limit of the latter reco vers the contin uous BSQ equation [49]; b oth the discrete BSQ equation and the p en tagram maps are gov erned b y third-order sp ectral problems in the Lax form ulation. On the other hand, a PGL(3)-inv ariant lattice BSQ system w as obtained as a reduction of a Q3 analogue of the lattice BSQ equations [61], though without explicit geometric considerations. Moreo ver, a remark able PGL(3)-inv ariant system of PDEs, kno wn as gener ating PDEs for the BSQ hierarc hy w as introduced in [62, 63] (see Section 4.3 for details); ho wev er, its pro jective form ulation has remained only a conjecture. The primary ob jective of the presen t w ork is to provide a uniﬁed approach to constru ct- ing PGL(3)-in v arian t integrable systems related to the BSQ equations. Our metho dology is grounded in linear sp ectral problems and their factorised forms. The con tinuous and discrete realms are connected through the notion of dualities induced b y the factorisation. This yields a uniﬁed represen tation of all systems in terms of a common set of dep enden t v ariables throughout the pap er (denoted as z 1 , z 2 that are the inhomogeneous co ordinates of the sp ectral problems), providing a coheren t algebraic and geometric picture across diﬀeren t in tegrable regimes. Our main results include: • Explicit formulae for the gener ating set of PGL(3)-diﬀeren tial and diﬀerence inv ari- an ts in the sense of [31, 46] (see also Remark 2.1). They are given in Deﬁnitions 2.2 and 2.8 generalising resp ectiv ely the Sc hw arzian deriv ativ e and cross-ratio. W e also pro vide c omp osition rules for the PGL(3)-diﬀerential inv ariants, c.f. Theorem 2.4, and contin uum limits of PGL(3)-diﬀerence inv ariants, c.f. Prop osition 2.10. • Con tinuous–discrete duality and self-duality induced by factorisation of linear dif- feren tial or diﬀerence op erators (see Section 3), underlying the exact discretisation and m ulti-dimensional consistency of the asso ciated BSQ systems; w e also pro vide the asso ciated Darb oux–Crum formulae (see App endix C). • Deriv ation of the contin uous and discrete PGL(3)-inv ariant BSQ systems, c.f. Sec- tions 4.1 and 4.2. A lifting-de c oupling me chanism is pro vided to reduce PGL(3)- in v arian t BSQ systems to (PGL(2)-inv arian t) Sc hw arzian BSQ equations. The lift 2 of the discrete PGL(3)-inv ariant BSQ systems is a three-component quad-system, consisten t around a cub e serving as the rank-3 analogue of the cross-ratio equation. • Deriv ation of the PGL(3) -invariant gener ating PDE —a coupled system of PDEs with the asso ciated lattice parameters serving as the indep enden t v ariables, con taining the entire hierarch y of PGL(3)-inv ariant BSQ equations by systematic expansions— together with its Lagrangian structure, c.f. Section 4.3, corresponding to the pro jec- tiv e form ulation of the BSQ generating systems [62, 63]. Our w ork establishes a uniﬁed pro jective formulation for PGL(3)-inv ariant integrable systems, p ositioning the rank-3 BSQ setting as the natural generalization of the w ell- understo od rank-2 KdV theory . The metho ds dev elop ed here could naturally extend to higher-rank cases oﬀering a systematic approach to PGL( N )-inv ariant integrable systems. Note that our constructions are mainly algebraic in nature and apply equally to systems deﬁned ov er the real ﬁeld R or the complex ﬁeld C . While the complex case is p erhaps the most relev ant, for the sake of notational simplicity , we shall omit explicit references to the base ﬁeld and use the general notations suc h as PGL(3) and P 2 throughout the paper. The pap er is organised as follows. Section 2 derives the generating PGL(3)-diﬀeren tial and diﬀerence inv ariants (see Remark 2.1 for the deﬁnition of generating inv ariants) from third-order spectral problems and establishes their related prop erties suc h as composition rules and contin uous limits. Section 3 studies factorisation of the third-order sp ectral problems, rev ealing the con tinuous–discrete duality and self-duality among the asso ciated BSQ equations. Sections 4.1 and 4.2 present the con tin uous, discrete PGL(3)-in v arian t BSQ systems together with their geometric interpretation and decoupling mechanism. Section 4.3 presents the PGL(3)-inv ariant generating PDE and its Lagrangian structure, with some details of the deriv ation given in App endix D. App endices A and B presen t the general theory of PGL( N )-in v arian ts for arbitrary N . App endix C pro vides the asso ciated Darb oux–Crum form ulae. 2 Pro jectiv e in v arian ts from linear sp ectral problems In this section, we pro vide explicit form ulae for PGL(3)-diﬀeren tial and diﬀerence inv ari- an ts derived from third-order linear sp ectral problems, along with the c omp osition rules for the PGL(3)-diﬀeren tial in v arian ts. W e also presen t their multiv ariate extensions, whic h will b e crucial for the developmen t of PGL(3)-inv ariant BSQ-t yp e in tegrable systems dis- cussed in subsequen t sections. A group-theoretical approac h to the in v ariants and a general framew ork for PGL( N )-inv arian ts will b e presented resp ectiv ely in App endices A and B. It is a classical result in pro jectiv e diﬀerential geometry that a scalar linear diﬀeren- tial equation of order N corresp onds to a curve in the pro jectiv e space P N − 1 [68]. The indep enden t solutions of the linear diﬀeren tial equation serve as homogeneous co ordinates of the lifted curve, and the co eﬃcients (or p oten tials), up to gauge transformations, are the generating diﬀerential pro jectiv e inv ariants. Although this geometric corresp ondence is well established in the con text of classical inv ariant theory , conformal ﬁeld theory and in tegrable systems (see, for example, [10, 11, 16, 50, 51, 68]), our fo cus is to pro vide explicit deriv ations of these in v arian ts, and extend this framework to the discrete realm, where generating diﬀerence in v arian ts arise from linear diﬀerence equations of order N corre- sp onding to discrete pro jective curves (p olygons) in P N − 1 . Belo w, w e fo cus on the N = 2 and N = 3 cases, relev an t to our fo cus on BSQ-t yp e equations. 3 2.1 Pro jectiv e diﬀerential inv arian ts W e ﬁrst illustrate the well-kno wn metho d for constructing the PGL(2)-diﬀerential in v ari- an t, namely the Sch warzian deriv ative, based on a second-order linear ordinary diﬀeren tial equation. Then, w e apply the same tec hnique to a third-order equation, and deriv e the PGL(3)-diﬀeren tial in v arian ts that are higher-order analogs of the Sch w arzian deriv ativ e. This pro cedure can be generalized as a systematic algorithm to obtain PGL( N )-diﬀerential in v arian ts based on an N th-order linear ordinary diﬀerential equation (see App endix B.1 for further details). 2.1.1 PGL(2) -diﬀeren tial inv arian t and the Sc hw arzian deriv ativ e Consider the Sc hr¨ odinger sp ectral problem ( ∂ 2 x + u ) φ = λφ . (2.1) The p oten tial u is supp osed to b e a smo oth function, and λ plays the role of the sp ectral parameter. This equation is also known as the Liouvil le normal form , as a general second- order linear ordinary diﬀeren tial equation that can alw a ys be transformed to (2.1) [68]. F rom the viewp oin t of ordinary diﬀerential equations, the sp ectral parameter λ could b e absorb ed into u . Ho wev er, in the context of in tegrable systems, the app earance of λ is related to the imp ortan t notion of Lax pair. Therefore, we k eep using the spectral problem (2.1) in this pap er. The construction of the in v ariant is based on the observ ation that (2.1) has t wo linearly indep enden t solutions, sa y φ 1 , φ 2 , and that the general solution is a linear com bination of those tw o solutions: φ = c 1 φ 1 + c 2 φ 2 . Introduce the inhomogeneous co ordinate z as the ratio of φ 1 , φ 2 , i.e. z := φ 1 /φ 2 . Geometrically , z represen ts an aﬃne co ordinate of a class of nondegenerate pro jectively equiv alent curv es in P 1 (nondegeneracy means z x  = 0) [50, 68]. F or simplicity , w e set φ 2 = φ , and rewrite (2.1) for φ 1 = z φ as z xx φ + 2 z x φ x = 0 . (2.2) This leads to ∂ x log φ = − z xx 2 z x , (2.3) where the left-hand side is a Hopf–Cole transformation for φ . Substituting the ab o ve form ula in to the spectral problem (2.1) for φ to eliminate φ in fav our of z , w e can express u in terms of z as follo ws 2( u − λ ) = S [ z ] , S [ z ] := z xxx z x − 3 2 z 2 xx z 2 x . (2.4) The expression S [ z ], (or S [ z ]( x ) if the indep endent v ariable needs to b e sp eciﬁed), is the Schwarzian derivative . The quantit y u − λ is inv ariant under the action of GL(2) on the indep enden t solutions ( φ 1 , φ 2 ), namely ( φ 1 , φ 2 ) 7→ ( φ 1 , φ 2 ) M , M =  m 11 m 12 m 21 m 22  , det M  = 0 . (2.5) This induces a PGL(2)-action on z : z 7→ m 11 z + m 21 m 12 z + m 22 , (2.6) under which S [ z ] is inv ariant. 4 Remark 2.1 F or a Lie gr oup acting r e gularly on a jet sp ac e, ther e exists a ﬁnite gen- er ating set of diﬀer ential invariants, such that every diﬀer ential invariant c an b e lo c al ly written as a function of the gener ating invariants and their derivatives using invariant dif- fer ential op er ators (se e, e.g. [31, 46]). Although the PGL -actions ar e not ne c essary r e gular themselves, they b e c ome r e gular when they ar e pr olonge d to acting on jet sp ac es. In the PGL(2) -action (2.6) for ( x, z ) , the gener ating invariant is the Schwarzian derivative S [ z ] ; every diﬀer ential invariant c an b e expr esse d as a function of S [ z ] and its derivatives with r esp e ct to x . A n alternative derivation of the Schwarzian derivative S [ z ] using inﬁnitesimal gener ators is pr ovide d in App endix A.1. 2.1.2 PGL(3) -diﬀeren tial inv arian ts Consider the follo wing third-order sp ectral problem ( ∂ 3 x + u∂ x + v ) φ = λφ , (2.7) and let φ 1 , φ 2 , φ 3 b e the set of linearly indep enden t solutions. The natural GL(3) symmetry on the solutions ( φ 1 , φ 2 , φ 3 ) 7→ ( φ 1 , φ 2 , φ 3 ) M , M = ( m ij ) , det M  = 0 , (2.8) induces a PGL(3) transformation (a righ t action) on the inhomogeneous co ordinates z 1 = φ 1 φ 3 , z 2 = φ 2 φ 3 , (2.9) namely , ( z 1 , z 2 ) 7→  m 11 z 1 + m 21 z 2 + m 31 m 13 z 1 + m 23 z 2 + m 33 , m 12 z 1 + m 22 z 2 + m 32 m 13 z 1 + m 23 z 2 + m 33  . (2.10) Geometrically , ( z 1 , z 2 ) serv e as aﬃne co ordinates of a class of nondegenerate pro jectively equiv alen t curves in P 2 (nondegeneracy means ∂ 2 x z 1 ∂ x z 2 − ∂ 2 x z 2 ∂ x z 1  = 0). Next, w e construct the generating diﬀeren tial in v arian ts in a similar w a y as the PGL(2) case. W riting the equations for φ = φ 3 and φ i = z i φ , i = 1 , 2, we obtain the following system for both z 1 and z 2 : − u = z (3) i z (1) i + 3 z (2) i z (1) i ∂ x log φ + 3  ∂ 2 x log φ + ( ∂ x log φ ) 2  , (2.11a) λ − v = − ( ∂ x log φ ) " z (3) i z (1) i + 3 z (2) i z (1) i ∂ x log φ # + ∂ 2 x log φ − 2( ∂ x log φ ) 3 , (2.11b) where z ( j ) i := ∂ j x z i . F rom (2.11a) for b oth i = 1 , 2, w e c an eliminate u and express ∂ x log φ as a ratio of tw o determinants ∂ x log φ = − 1 3   z (3) , z (1)     z (2) , z (1)   , (2.12) where z = ( z 1 , z 2 ) ⊺ . This is then substituted bac k to (2.11) and we obtain u and v − λ in 5 terms of z 1 and z 2 , which turns out to b e symmetric with resp ect to z 1 and z 2 : u =   z (4) , z (1)     z (2) , z (1)   + 2   z (3) , z (2)     z (2) , z (1)   − 4 3   z (3) , z (1)     z (2) , z (1)   ! 2 , (2.13a) v − λ = 1 3 u x − 8 27   z (3) , z (1)     z (2) , z (1)   ! 3 − 1 3   z (4) , z (2)     z (2) , z (1)   (2.13b) + 8 9   z (3) , z (2)     z (3) , z (1)     z (2) , z (1)   2 + 2 9   z (4) , z (1)     z (3) , z (1)     z (2) , z (1)   2 . Deﬁnition 2.2 L et   z (2) , z (1)    = 0 . We deﬁne the quantities S 1 [ z 1 , z 2 ] and S 2 [ z 1 , z 2 ] as S 1 [ z 1 , z 2 ] =   z (4) , z (1)     z (2) , z (1)   + 2   z (3) , z (2)     z (2) , z (1)   − 4 3   z (3) , z (1)     z (2) , z (1)   ! 2 , (2.14a) S 2 [ z 1 , z 2 ] =   z (4) , z (2)     z (2) , z (1)   − 2 3   z (4) , z (1)     z (3) , z (1)     z (2) , z (1)   2 (2.14b) − 8 3   z (3) , z (2)     z (3) , z (1)     z (2) , z (1)   2 + 8 9   z (3) , z (1)     z (2) , z (1)   ! 3 . Here, S 1 [ z 1 , z 2 ] , S 2 [ z 1 , z 2 ] are related to the p oten tials u, v as S 1 [ z 1 , z 2 ] = u , S 2 [ z 1 , z 2 ] = − 3( v − λ ) + u x , (2.15) whic h are clearly PGL(3)-inv ariant. They can b e used as the set of PGL(3) gener ating diﬀer ential invariants of a single indep enden t v ariable, c.f. Remark 2.1. Remark 2.3 Sinc e al l diﬀer ential invariants c onstructe d in this way ar e r ational functions in the jet sp ac e, we c an deﬁne their weight as the de gr e e of homo geneity in terms of derivatives. Sp e ciﬁc al ly, the weight is the total numb er of derivatives in the numer ator minus the total numb er of derivatives in the denominator. In this sense, S 1 has a weight of 2 , while S 2 has a weight of 3 ; the Schwarzian derivative S [ z ] has weight 2 . 2.2 Comp osition rules for the diﬀeren tial in v arian ts The composition rule (also known as connection formula) of Sc hw arzian deriv ativ e is a result of smooth deformation of the second-order linear equation (2.1), and has close con- nection to imp ortan t notions of mathematical physics suc h as Virasoro algebras, KdV hierarc hy and conformal ﬁeld theory [10, 11, 16, 50, 51]. Here, we pro vide its explicit deriv a- tion and generalise it to the PGL(3)-inv ariants S 1 [ z 1 , z 2 ] , S 2 [ z 1 , z 2 ], c.f. (2.14). Consider the Sc h warzian deriv ativ e. Let y = f ( x ) b e a smooth function and deﬁne φ ( x ), satisfying (2.1), as a comp osite φ ( x ) = ψ ( y ) = ψ ◦ f ( x ). Substituting it back to the sp ectral problem yields a linear equation for ψ ( y ): ∂ 2 y ψ + W ( y ) ∂ y ψ + U ( y ) ψ = 0 , (2.16) where W ( y ) = f xx f 2 x , U ( y ) = u − λ f 2 x = S [ z ] 2 f 2 x . (2.17) 6 Starting with tw o indep enden t solutions ψ 1 ( y ) and ψ 2 ( y ), we deﬁne their ratio as Z ( y ) = ψ 1 ( y ) /ψ 2 ( y ); therefore, the tw o independent solutions of the original sp ectral problem are φ 1 ( x ) = ψ 1 ◦ f ( x ) , φ 2 ( x ) = ψ 2 ◦ f ( x ) , (2.18) and hence their ratio is z = φ 1 φ 2 = ψ 1 ψ 2 ◦ f = Z ◦ f . (2.19) Substituting ψ 1 = Z ψ with ψ = ψ 2 to (2.16) yields ∂ y log ψ = − 1 2 Z y y Z y − W 2 . (2.20) This is substituted back to (2.16) and w e get the following iden tity S [ Z ] = 2 U − ∂ y W − 1 2 W 2 . (2.21) By using the chain rule ∂ y W = ∂ x W ∂ x y = ∂ x W f x , (2.22) and taking (2.17) into accoun t, the identit y (2.21) b ecomes S [ Z ◦ f ] = f 2 x ( S [ Z ] ◦ f ) + S [ f ] , (2.23) whic h is the c omp osition rule of the Schwarzian derivative . No w consider the PGL(3)-diﬀerential inv ariants S 1 [ z 1 , z 2 ] , S 2 [ z 1 , z 2 ]. Similarly , let y = f ( x ) be a smo oth function and deﬁne φ ( x ) = ψ ( y ) = ψ ◦ f ( x ). Inserting this to (2.7) yields ∂ 3 y ψ + W ( y ) ∂ 2 y ψ + U ( y ) ∂ y ψ + V ( y ) ψ = 0 , (2.24) where W ( y ) = 3 f xx f 2 x , U ( y ) = f xxx + uf x f 3 x , V ( y ) = v − λ f 3 x . (2.25) Let ψ 1 ( y ), ψ 2 ( y ) and ψ 3 ( y ) b e three indep enden t solutions of (2.24) and deﬁne Z 1 = ψ 1 ψ 3 , Z 2 = ψ 2 ψ 3 . (2.26) This gives three indep endent solutions of the original sp ectral problem, i.e. φ i = ψ i ◦ f ( i = 1 , 2 , 3) and hence their ratios are z 1 = φ 1 φ 3 = Z 1 ◦ f , z 2 = φ 2 φ 3 = Z 2 ◦ f . (2.27) Let ψ = ψ 3 . The equations for ψ 1 = Z 1 ψ and ψ 2 = Z 2 ψ lead to ∂ y log ψ = − 1 3    Z (3) , Z (1)       Z (2) , Z (1)    − W 3 . (2.28) Substituting this bac k to (2.24), we obtain the follo wing tw o identities S 1 [ Z 1 , Z 2 ] = U − ∂ y W − 1 3 W 2 , (2.29a) S 2 [ Z 1 , Z 2 ] = − 3 V + ∂ y S 1 [ Z 1 , Z 2 ] + W S 1 [ Z 1 , Z 2 ] + ∂ 2 y W + W ∂ y W + 1 9 W 3 . (2.29b) T aking the functions W ( y ), U ( y ) and V ( y ) in to consideration, the iden tities (2.29) can b e rewritten. W e summarize the ﬁnal result as the follo wing theorem. 7 Theorem 2.4 The c omp osition rules of the PGL(3) -diﬀer ential invariants (2.14) ar e S 1 [ Z 1 ◦ f , Z 2 ◦ f ] = f 2 x ( S 1 [ Z 1 , Z 2 ] ◦ f ) + 2 S [ f ] , (2.30a) S 2 [ Z 1 ◦ f , Z 2 ◦ f ] = f 3 x ( S 2 [ Z 1 , Z 2 ] ◦ f ) − f x f xx ( S 1 [ Z 1 , Z 2 ] ◦ f ) − ∂ x S [ f ] , (2.30b) wher e S [ f ] is the Schwarzian derivative. Remark 2.5 The ab ove formulae enc o de the tr ansformation pr op erties of PGL(3) -diﬀer ential invariants under r ep ar ametrizations x 7→ y = f ( x ) , and wer e c entr al to classic al W n ( n = 3 her e) algebr a [10, 11, 16, 51]. Our emphasis is put on explicit formulae in terms of the in- homo gene ous c o or dinates z 1 , z 2 . In W 3 algebr a, S 1 [ z 1 , z 2 ] c orr esp onds to a quasi-primary ﬁeld of weight 2 (whose inﬁnitesimal version c orr esp onds to the Vir asor o algebr a), while an alternative invariant S 2 [ z 1 , z 2 ] = − 1 6 ∂ x S 1 [ z 1 , z 2 ] − 1 3 S 2 [ z 1 , z 2 ] , (2.31) r esulte d fr om S 2 [ z 1 , z 2 ] = v − λ − u x 2 , tr ansforms as a genuine weight 3 diﬀer ential [16]: S 2 [ Z 1 ◦ f , Z 2 ◦ f ] = f 3 x ( S 2 [ Z 1 , Z 2 ] ◦ f ) . (2.32) Then, S 1 [ z 1 , z 2 ] and S 2 [ z 1 , z 2 ] c ould form another set of gener ating PGL(3) -diﬀer ential invariants. Ge ometric al ly, S 1 [ z 1 , z 2 ] c oincides with the pro jective curv ature of a nonde- gener ate curve, while S 2 [ z 1 , z 2 ] governs the so-c al le d pro jectiv e arc-length element [50]. 2.3 Diﬀerence pro jectiv e inv arian ts The technique of constructing diﬀerential inv arian ts can b e well adapted to the discrete realm. Here, we consider functions dep ending on a discrete v ariable, sa y n ∈ Z . W e emplo y T as the forw ard shift op erator, and use the ¯ notations to denote shifts. F or a function φ := φ ( n ), one has the forw ard shifts as T φ = φ = φ ( n + 1) , T 2 φ = φ = φ ( n + 2) , . . . , (2.33) and the bac kward shifts as T − 1 φ = φ = φ ( n − 1) , . . . (2.34) 2.3.1 The PGL(2) -diﬀerence inv arian t: cross-ratio Consider a second-order linear diﬀerence equation ( T 2 + h T + α ) φ = λφ . (2.35) Here φ and h are functions of n ∈ Z , α is the lattice parameter and λ is the sp ectral parameter. Let z b e the ratio of t wo linearly indep enden t solutions φ 1 , φ 2 of (2.35): z = φ 1 /φ 2 . The action of GL(2) giv en b y (2.5) on the solution space of the linear problem (2.35) induces fractional linear transformations to z as shown in (2.6). Let φ 2 = φ , and the linear ordinary diﬀerence equation (2.35) gives rise to the system of equations: φ + h φ = ( λ − α ) φ , (2.36a) z φ + h z φ = ( λ − α ) z φ , (2.36b) 8 whic h w e can write as:  1 1 z z  φ/φ h φ/φ ! = ( λ − α )  1 z  . (2.37) Solving this system yields φ φ = ( λ − α )     1 1 z z         1 1 z z     , h φ φ = ( λ − α )     1 1 z z         1 1 z z     , (2.38) from which by eliminating φ (by shifting the second forward, multiplying it b y its original form, and equating the result with the ﬁrst expression), we obtain the equalit y: − hh λ − α =  z − z   z − z   z − z  ( z − z ) . (2.39) The right-hand side, as a cr oss-r atio of four p oin ts z , z , z , z , is manifestly PGL(2)- in v arian t. Moreov er, it is a gener ating PGL(2) -diﬀer enc e invariant such that every diﬀer- ence in v arian t can b e expressed as a function of it and its shifts (see App endix A.2 for a pro of, and Remark 2.1 for the diﬀeren tial coun terpart). Remark 2.6 In the c ontinuous c ase, the set of gener ating diﬀer ential invariant is essen- tial ly unique (up to functional dep endenc e). F or instanc e, for the PGL(2) -action (2.6) , the only functional ly indep endent thir d-or der diﬀer ential invariant is the Schwarzian derivative (2.4) . In c ontr ast, in the discr ete setting the set of gener ating diﬀer enc e invariant is not unique, sinc e it dep ends on the choic e of stencil. F or example, the gener ating diﬀer enc e invariant (2.39) is deﬁne d in terms of z , z , z , z , but an e qual ly valid invariant c an b e deﬁne d on shifte d stencils such as z , z , z , z ¯ , or other four-p oint c onﬁgur ations. These invariants ar e r elate d by lattic e shifts and ther efor e gener ate the same algebr a of diﬀer enc e invariants. Remark 2.7 Ge ometric al ly, (2.35) deﬁnes a class of pr oje ctively e quivalent nonde gener- ate discr ete pr oje ctive curves (or p olygons) in P 1 . The nonde gener acy me ans that any 2 c onse cutive vertic es sp an the ful l pr oje ctive sp ac e P 1 , i.e. z − z  = 0 . This notion c an b e gener alize d to the c orr esp ondenc e b etwe en an N th-or der line ar diﬀer enc e e quation and discr ete pr oje ctive curves in P N − 1 [49]. One can take straigh tforwardly con tinuum limits b y expanding the shifted ob jects through T aylor expansions to obtain the corresp onding diﬀerential in v arian ts. F or instance, taking z = z ( x ) = z ( x 0 + nδ ) , z = z ( x + δ ) = z + δ z x + 1 2 δ 2 z xx + · · · , (2.40) Expanding (2.39) in p ow ers of δ yields  z − z   z − z   z − z  ( z − z ) = 4  1 − 1 2 δ 2 S ( z ) + · · ·  , (2.41) th us reco vering the Sch w arzian deriv ativ e. 9 2.3.2 The PGL(3) -diﬀerence inv arian ts W e now consider a third-order linear diﬀerence equation Λ φ = λφ , Λ = T 3 + h T 2 + g T + α . (2.42) W e hav e the same GL(3)-action as in the con tinuous case given by (2.8) on the space of solutions. In tro ducing again z 1 = φ 1 /φ 3 and z 2 = φ 2 /φ 3 and letting φ = φ 3 , we get the follo wing set of equations: φ + h φ + g φ = ( λ − α ) φ , (2.43a) z i φ + h z i φ + g z i φ = ( λ − α ) z i φ , ( i = 1 , 2) , (2.43b) whic h w e can write as:    1 1 1 z 1 z 1 z 1 z 2 z 2 z 2       φ/φ h φ/φ g φ/φ    = ( λ − α )   1 z 1 z 2   . (2.44) Solving the system using Cramer’s rule we obtain the following expressions: φ φ = ( λ − α )   z , z , z      z , z , z    , φ φ = − λ − α h    z , z , z       z , z , z    , φ φ = λ − α g    z , z , z       z , z , z    , (2.45) where z denotes a 3-comp onent v ector z = (1 , z 1 , z 2 ) ⊺ 1 , and the numerators and denomi- nators in the ab o v e expressions b eing 3 × 3 determinan ts consisting of these v ectors with the asso ciated shifts applied to them. Eliminating φ , w e obtain t wo indep enden t inv ariants: − gg ( λ − α ) h =    z , z , z       z , z , z       z , z , z       z , z , z    , − hg λ − α =    z , z , z       z , z , z       z , z , z      z , z , z   . (2.46) These can be considered as the generating diﬀerence inv ariants corresp onding to the PGL(3)-action in the discrete case; see App endix A.2 for more details. Deﬁnition 2.8 L et   z , z , z    = 0 . We deﬁne the quantities I 1 [ z 1 , z 2 ] and I 2 [ z 1 , z 2 ] as I 1 [ z 1 , z 2 ] =    z , z , z       z , z , z       z , z , z       z , z , z    , I 2 [ z 1 , z 2 ] =    z , z , z       z , z , z       z , z , z      z , z , z   . (2.47) The I 1 [ z 1 , z 2 ] , I 2 [ z 1 , z 2 ] can b e chosen as the gener ating PGL(3) -diﬀer enc e invariants . It should b e noted that the generating in v arian ts ma y hav e v arious equiv alen t expressions due to the Pl ¨ uc ker relation for 3D v ectors as follows | a , b , d | | c , d , f | = | a , c , d | | b , d , f | − | b , c , d | | a , d , f | . (2.48) 1 W e retain the notation z , although its dimension diﬀers from that in Section 2.1.2 where z denoted a 2-comp onent v ector ( z 1 , z 2 ) ⊺ . This distinction is apparent from the determinan tal structures inv olved (2 × 2 versus 3 × 3 determinants). 10 Remark 2.9 In c ontr ast to the c ontinuous c ase wher e the p otentials in the line ar diﬀer- ential op er ators ar e r ational expr essions of the inhomo gene ous c o or dinates (se e r esp e ctively (2.4) and (2.13) for the PGL(2) and PGL(3) c ases), one has quadr atic c ombinations of the discr ete p otentials as r ational expr essions of the inhomo gene ous c o or dinates, c.f. (2.39) and (2.46) . F r om a diﬀer ential/diﬀer enc e Galois the ory p ersp e ctive, the diﬀer enc e pr oje ctive invariants ar e inde e d quadr atic extension of the diﬀer enc e p otentials (se e, e.g. [36]). Analogous to the cross-ratio’s con tinuum limit, the generating PGL(3)-diﬀerence in- v arian ts (2.47) admit the following con tinuum limits. Prop osition 2.10 L et I 1 [ z 1 , z 2 ] , I 2 [ z 1 , z 2 ] b e the PGL(3) -diﬀer enc e invariants given by (2.47) , and let z 1 , z 2 b e smo oth functions of x = x 0 + nδ , wher e n is the discr ete variable and δ is an inﬁnitesimal quantity, i.e. z 1 := z 1 ( x ) = z 1 ( x 0 + nδ ) , z 2 := z 2 ( x ) = z 2 ( x 0 + nδ ) . (2.49) Then, by p erforming T aylor exp ansion in δ , one gets I 1 [ z 1 , z 2 ] = 3 − S 1 [ z 1 , z 2 ] δ 2 −  2 ∂ x S 1 [ z 1 , z 2 ] + 1 2 S 2 [ z 1 , z 2 ]  δ 3 + O ( δ 4 ) , (2.50a) I 2 [ z 1 , z 2 ] = 9 − 6 S 1 [ z 1 , z 2 ] δ 2 − 12 ∂ x S 1 [ z 1 , z 2 ] δ 3 + O ( δ 4 ) . (2.50b) Pro of: This relies on series expansion of I 1 [ z 1 , z 2 ] , I 2 [ z 1 , z 2 ] in δ . By iden tifying the highest- order deriv ativ es in the co eﬃcien ts of the expansions, one can identify those co eﬃcien ts with the diﬀeren tial inv ariants S 1 [ z 1 , z 2 ] and S 2 [ z 1 , z 2 ]. Remark 2.11 The thir d-or der line ar diﬀer enc e e quation (2.42) was use d in the L ax for- mulation of p entagr am maps [24, 49, 60], wher e the c o eﬃcients (discr ete p otentials) h and g wer e employe d as dual c o or dinates to r epr esent the maps. However, an explicit pr oje ctive formulation of the invariants in terms of the inhomo gene ous c o or dinates z 1 , z 2 was not pr ovide d in that c ontext. It is also inter esting to note that these PGL(3) -invariants, as a gener alisation of the cr oss-r atio, have b e en develop e d indep endently in the c ommunity of image r e c o gnition and c omputer vision; se e, e.g. [37, 65]. 2.4 P artial diﬀerential & diﬀerence in v ariants The in v arian t deriv ed in the previous sections for a single independent v ariable can b e nat- urally extended to a t wo-v ariable setting—that is, from curv es to surfaces. Suc h extensions are essen tial for constructing pro jective-in v ariant in tegrable systems (see Section 4). Here, w e state the results for PGL(2)- and PGL(3)-inv ariants with t wo indep enden t v ariables, but omit the details, as they can b e deriv ed through systematic metho ds. These include the use of inﬁnitesimal generators (App endix A; see, also, [22, 47]), symbolic computations ( e.g. [20, 21, 27]), or mo ving frames ( e.g. [13, 31–33, 67]). P artial diﬀeren tial inv arian ts. In addition to the x , let us introduce an extra inde- p enden t v ariable t . • PGL(2)-action on ( x, t, z ). A new generating in v arian t is z t z x . (2.51) 11 The inv ariant z t /z x and the Sch w arzian deriv ativ e S [ z ] giv en by (2.4) form the set of generating inv ariants for the PGL(2)-action with t wo indep enden t v ariables and one dep enden t v ariable. They generate all other PGL(2)-diﬀeren tial in v arian ts as functions of them as well as their deriv ativ es with resp ect to x and t . • PGL(3)-action on ( x, t, z 1 , z 2 ). The new generating inv ariants can b e c hosen R 1 =   z t , z (1)     z (2) , z (1)   , R 2 =   z t , z (2)     z t , z (1)   − 2 3   z (3) , z (1)     z (2) , z (1)   , (2.52) where z = ( z 1 , z 2 ) ⊺ , and z ( j ) denotes j -th deriv ativ es with respect to x . Similarly , to- gether with the in v ariants S 1 [ z 1 , z 2 ] , S 2 [ z 1 , z 2 ] deﬁned in Deﬁnition 2.2, they generate all PGL(3)-diﬀeren tial inv ariants with t wo independent v ariables and tw o dep enden t v ariables. P artial diﬀerence in v ariants. The deriv ation of diﬀerence inv ariants for multiple in- dep enden t v ariables is more straigh tforward than in the diﬀeren tial case. It simply in v olves replacing certain v ariables with their shifted coun terparts in the new direction, a fact that follo ws directly from the inﬁnitesimal inv ariance condition (see App endix A.2). This is generalisable to PGL( N ) ( N > 3) case: new generating diﬀerence inv ariants when a sec- ond discrete v ariable m is introduced can b e obtained by replacing all N -th forw ard shifts z ( N ) b y e z in (B.25). Let us in tro duce a new discrete v ariable, say m , asso ciated to a e shift op erator. • PGL(2)-action on ( n, m, z ). A new generating inv ariant can b e c hosen as, for in- stance, one of the following in v arian ts: ( e z − z ¯ )( z − z ) ( e z − z )( z − z ¯ ) , ( e z − z )( z − z ) ( e z − z )( z − z ) , ( e z − z )( z − z ¯ ) ( e z − z )( z − z ¯ ) , ( e z − z )( z − z ) ( e z − z )( z − z ) . (2.53) These inv ariants are connected through the determinan tal relation for 2-comp onen t v ectors | a , b | | c , d | = | a , c | | b , d | − | a , d | | b , c | . (2.54) If we consider the ﬁrst-order T a ylor expansions for the shifts z = z ( x + δ, t ) = z + δ z x + · · · , e z = z ( x, t + ε ) = z + εz t + · · · , (2.55) the ﬁrst in v arian t in (2.53), for instance, b ecomes ( e z − z ¯ )( z − z ) ( e z − z )( z − z ¯ ) = 1 2  1 + δ ε z x z t + · · ·  , (2.56) giving the diﬀeren tial inv ariant z t /z x . • PGL(3)-action on ( n, m, z 1 , z 2 ). W e can immediately obtain the new generating diﬀerence inv ariants, for instance, by replacing all the z with e z in (2.46):    e z , z , z       z , z , z       e z , z , z       z , z , z    ,   e z , z , z      z , z , z       e z , z , z      z , z , z   , (2.57) where z = (1 , z 1 , z 2 ) ⊺ . 12 Similarly to the diﬀeren tial case, these new diﬀerence inv ariants, together with those obtained in Section 2.3, constitute the corresp onding generating set of diﬀerence in v ari- an ts, from whic h all diﬀerence in v arian ts can be expressed as functions of these inv ariants and their shifts. F or instance, the cross-ratio (2.39) and any one of (2.53) form the gen- erating inv ariants of the PGL(2)-action with t wo discrete indep enden t v ariables and one dep enden t v ariable, while the inv arian ts (2.47) and (2.57) form the generating inv ariants of the PGL(3)-action with tw o discrete indep enden t v ariables and tw o dep enden t v ariables. Diﬀeren tial-diﬀerence in v ariants. Now, let t b e the con tinuous indep endent v ariable and n be the discrete one asso ciated to the shift ¯ . • PGL(2)-action on ( t, n, z ). The new generating diﬀerential-diﬀerence inv ariant can b e c hosen as ( z − z ¯ ) ( z − z ¯ )( z − z ) z t . (2.58) • PGL(3)-action on ( t, n, z 1 , z 2 ). The new generating inv ariants can b e c hosen as    z t , z , z       z , z , z       z t , z , z       z , z , z    ,   z t , z , z      z , z , z       z t , z , z      z , z , z   . (2.59) Again, these new diﬀerential-diﬀerence inv ariants, together with the diﬀerence in v ari- an ts obtained in Section 2.3, can b e c hosen as generating inv ariants: the cross-ratio (2.39) and the diﬀerential-diﬀerence inv ariant (2.58) form a generating set for the PGL(3)-action with tw o indep endent v ariables, one con tinuous and one discrete, and one dep enden t v ari- able, while the diﬀerence inv ariants (2.47) and the diﬀeren tial-diﬀerence in v arian ts (2.59) form a generating set for the PGL(3) action with t wo indep enden t v ariables, one con tinu- ous and one discrete, and t wo dep enden t v ariables; see [67] for more examples. It should also b e noted that the diﬀeren tial-diﬀerence in v arian t theory is not merely a simple combi- nation of diﬀerential and diﬀerence theories (see e.g. [52, 53]); rather, it requires additional conditions that reﬂect the prolongation structure, and a comprehensiv e general theory has y et to b e developed. Analogous to the con tinuum limits established in Prop osition (2.10), the partial dif- ference and semi-discrete in v arian ts introduced ab o v e admit limits that recov er the corre- sp onding m ulti-v ariable diﬀerential inv ariants. A detailed analysis is b ey ond the scop e of the present w ork. 3 F actorisation, dualit y , and exact discretisation of third- order sp ectral problems In this section, w e explore the in terplay betw een con tinuous and discrete third-order linear sp ectral problems through factorisation of op erators and Darb oux transformations. Start- ing from the con tin uous sp ectral problem (2.7), the associated Darb oux transformation in- duces a third-order discrete sp ectral problem (2.42) and giv es rise to a c ontinuous–discr ete duality , where the discrete problem constitutes an exact discr etisation of the contin uous one in the sense of Shabat [58, 59]. One rep eats this pro cedure in the discrete setting b y applying a discrete Darb oux transformation to the discrete sp ectral problem. This induces another cov ariant discrete op erator in a second discrete direction and leads to a 13 self-duality for the discrete op erators. Finally , we present the underlying nonlinear inte- grable BSQ equations expressed in terms of the PGL(3) inv arian ts introduced in Section 2. The new ingredient is that we also consider deformation of the discrete op erator (2.42) with resp ect to the lattice parameter yielding a non-autonomous semi-discrete BSQ sys- tem enco ding the complete hierarc hy of semi-discrete BSQ equations [62, 63]. Extensions to PGL(3)-inv ariant BSQ-t yp e systems will b e dev elop ed subsequently . Notation conv en tion: In Section 2, (partial) deriv atives were denoted using subscripts, e.g. z x , z t . In this Section and Sections 4.1-4.2, when considering contin uous BSQ-type PDEs, one uses primes for deriv atives with respect to the space v ariable and dots for deriv ativ es with resp ect to the time v ariable. The subscripts are reserved to present deriv ativ es with resp ect to lattice parameters, c.f. Section 4.3. 3.1 F rom contin uous to discrete: exact discretisation via Darb oux trans- formation The factorisation of linear diﬀeren tial operators w as initially studied in the con text of Hamiltonian structures and Miura transformations for integrable PDEs [1, 14, 15]. This framew ork allows for constructions of Darb oux transformations, and enables the passage from the contin uous sp ectral problem (2.7) to its discrete coun terpart (2.42) through the exact discretisation procedure [58, 59]. Assume L admits a factorised form L := ∂ 3 x + u∂ x + v = ( ∂ x − r 3 )( ∂ x − r 2 )( ∂ x − r 1 ) + α , (3.1) where α is a Darb oux–B¨ ac klund parameter. Expanding this yields the constraints 0 = r 1 + r 2 + r 3 , (3.2a) u = r 1 r 2 + r 1 r 3 + r 2 r 3 − r ′ 2 − 2 r ′ 1 , (3.2b) v = r ′ 1 r 2 + r 1 r ′ 2 + r ′ 1 r 3 − r 1 r 2 r 3 − r ′′ 1 + α . (3.2c) Eliminating r 3 = − r 1 − r 2 giv es a Miura-type transformation: u = − r 2 1 − r 2 2 − r 1 r 2 − 2 r ′ 1 − r ′ 2 , (3.3a) v = r 1 r 2 ( r 1 + r 2 ) − r 1 ( r ′ 1 − r ′ 2 ) − r ′′ 1 + α . (3.3b) Let φ 1 b e a ﬁxed solution of Lφ = λφ at λ = α , and deﬁne r 1 = (log φ 1 ) ′ . Substituting this into the sp ectral problem at λ = α yields a second-order Riccati-type equation r ′′ 1 + 3 r 1 r ′ 1 + ur 1 + r 3 1 + v = α . (3.4) A one-step Darb oux transformation is obtained by cyclically p erm uting the factors in (3.1), leading to the map in the follo wing lemma. Lemma 3.1 (One-step Darb oux transformation) L et φ 1 b e a ﬁxe d solution of Lφ = λφ at λ = α , and let r 1 = (log φ 1 ) ′ . Then the tr ansformation { φ, u, v } 7→ { φ, u, v } : φ = φ ′ − r 1 φ , u = u + 3 r ′ 1 , v = v + u ′ + 3( r 1 r ′ 1 + r ′′ 1 ) , (3.5) maps Lφ = λφ to Lφ = λφ wher e L = ∂ 3 x + u∂ x + v . 14 This follows from reordering the factors as L = ( ∂ x − r 1 )( ∂ x − r 3 )( ∂ x − r 2 ) + α . The action of iterated Darb oux transformations leads to compact forms for m ulti-soliton-type solutions, kno wn as Darb oux–Crum formulae. They will b e prese n ted in Appendix C. Note that Darb oux transformations for third-order sp ectral problems were considered previously in [4, 29], but from a diﬀerent persp ective than the one adopted here. Remark 3.2 Alternatively, one has another one-step Darb oux tr ansformation by letting b L = ( ∂ x − r 2 )( ∂ x − r 1 )( ∂ x − r 3 ) + α , b φ = ( ∂ x − r 2 )( ∂ x − r 1 ) φ , (3.6) which amounts to b u = u − 3 r ′ 3 , b v = v − u ′ − 3 r 3 r ′ 3 , (3.7) with b L = ∂ 3 x + b u∂ x + b v . One c an expr ess r 3 = − r 1 − r 2 with r 1 = (log φ 1 ) ′ , r 2 = (log φ 2 ) ′ , φ 2 = ( ∂ x − r 1 ) φ 2 . (3.8) Her e φ 1 , φ 2 ar e two line arly indep endent solutions of Lφ = λφ at λ = α . No w in terpret the Darb oux map as a shift op erator T acting on the eigenfunction: T φ := φ = ∂ x φ − r φ , (3.9) where w e suppress the subscript on r and allow it to dep end on the discrete v ariable n , i.e. r = r ( n, x ). Iterating this map yields φ = φ ′ − r φ , φ = φ ′ − r φ , φ = φ ′ − r φ . (3.10) Computing φ ′ using the ﬁrst tw o equations and φ ′ using the ﬁrst equation, one gets     φ φ φ φ     =     1 − r − r − r ∗ ⋆ 0 1 − r − r r r − r ′ 0 0 1 − r 0 0 0 1         φ ′′′ φ ′′ φ ′ φ     , (3.11) where ∗ = r ( r + r ) + r r − r ′ − 2 r ′ , ⋆ = − r r r + r r ′ + r ′ r + r r ′ − r ′′ . (3.12) Expressing φ, φ ′ , φ ′′ , φ ′′′ in terms of φ , φ , φ , φ , and using Lφ = λφ , w e obtain a third-order linear diﬀerence equation: Λ φ = φ + h φ + g φ + αφ = λφ , (3.13) where Λ = T 3 + h T 2 + g T + α with h = r + r + r , g = r 2 + r 2 + r r + r ′ + 2 r ′ + u . (3.14) Th us, the Darb oux iteration induces the discrete sp ectral problem (2.42). In versely , the linear problem (3.13) allo ws us to obtain the con tinuous sp ectral problem Lφ = λφ through the linear system (3.11). In this sense, the con tin uous op erator L and the discrete op erator Λ are dual: the former gov erns the x -evolution, and the latter the n -ev olution; they are connected by (3.9), and b oth share the same solution space spanned by φ 1 , φ 2 , φ 3 . 15 3.2 Discrete self-dualit y and multi-dimensional consistency Ha ving established exact discretisation in one lattice direction, w e no w apply the same factorisation tec hnique to the discrete op erator itself, whic h will generate a second lattice direction. Assume the discrete op erator Λ (3.13) admits a factorised form Λ := T 3 + h T 2 + g T + α = ( T − f 3 )( T − f 2 )( T − f 1 ) + β , (3.15) where β is another Darb oux–B¨ ac klund parameter, β  = α . Expanding this yields h = − f 3 − f 2 − f 1 = − f − f 1 , (3.16a) g = f 2 f 3 + f 1 f 3 + f 1 f 2 = f 2 f 3 + f 1 f , (3.16b) α = β − f 1 f 2 f 3 , (3.16c) where f := f 3 + f 2 . Eliminating f 2 f 3 = ( β − α ) /f 1 giv es a Miura-type transformation h = − f 1 − f , g = ( β − α ) /f 1 + f 1 f . (3.17) Let φ 1 b e a ﬁxed solution of Λ φ = λφ at λ = β , and deﬁne f 1 = φ 1 /φ 1 . Substituting it into the sp ectral problem at λ = β yields a second-order discrete Riccati-t yp e equation f 1 f 1 f 1 + h f 1 f 1 + g f 1 + α − β = 0 . (3.18) A one-step discrete Darboux transformation is obtained by cyclically permuting the factors in (3.15), leading to the follo wing theorem. Theorem 3.3 (One-step Darb oux transformation) L et φ 1 b e a ﬁxe d solution of Λ φ = λφ at λ = β , and let f 1 = φ 1 /φ 1 . Then the tr ansformation { φ, h , g } 7→ { e φ, e h , e g } : e φ = ( T − f 1 ) φ , e h = h + f 1 − f 1 , e g = g + f 1 h − f 1 h + f 1 ( f 1 + f 1 ) , (3.19) maps Λ φ = λφ to e Λ e φ = λ e φ wher e e Λ = T 3 + e h T 2 + e g T + α . The ab o ve theorem follo ws b y setting e Λ = ( T − f 1 )( T − f 3 )( T − f 2 ) + β . The action of iterated Darb oux transformations leads to compact forms kno wn as Darb oux–Crum form ulae. They will b e presented in App endix C. Remark 3.4 Alternatively, one has another one-step Darb oux tr ansformation by letting b Λ = ( T − f 2 )( T − f 1 )( T − f 3 ) + β , b φ = ( T − f 2 )( T − f 1 ) φ , (3.20) which amounts to b h = − f 2 − f 1 − f 3 , b g = f 1 f 2 + f 2 f 3 + f 1 f 3 , α = β − f 1 f 2 f 3 , (3.21) with b Λ = T 3 + b h T 2 + b g T + α . Comp aring to (3.16) , one has b h − h = f 3 − f 3 , b g − g = f 3 h − f 3 h − f 3 f 3 + f 3 f 3 . (3.22) In terpret the Darb oux map as a shift op erator S acting on the eigenfunction: S φ := e φ = ( T − f ) φ , (3.23) 16 where we suppress the subscript on f and allow it to dep end on the discrete v ariable m , i.e. f = f ( n, m ). Iterating this map yields e φ = φ − f φ , e e φ = e φ − e f e φ , e e e φ = e e φ − e e f e e φ . (3.24) As in (3.11), one obtains a third-order linear diﬀerence equation in the S shift: ( S 3 + p S 2 + q S + β ) φ = e e e φ + p e e φ + q e φ + β φ = λφ , (3.25) where p = f + e f + e e f + h , q = ( f + h )( f + e f ) + e f e f + g , β = ( f + h ) f f + f g + a . (3.26) No w compare the tw o sp ectral problems (3.15) and (3.25) connected b y (3.23). Intro- duce a potential v ariable w such that h = w − w , f = e w − w , (3.27) whic h is a result of e h = h + f − f in the Darb oux transformation (3.19). In terms of w , g b ecomes g = β − α f − ( f + h ) f = ( e w − w )( w − e w ) + β − α e w − w . (3.28) Remark ably , in terms of w , the co eﬃcien ts p and q in the induced spectral problem (3.25) tak e exactly the same forms as h and g resp ectively b y in terchanging { e , α } with { ¯ , β } . This rev eals a self-duality b et ween (3.15) and (3.25) through (3.23). The self-duality is enco ded in the symmetry: { T , α } ↔ { S, β } , (3.29) whic h exc hanges the roles of the tw o lattice directions and their asso ciated parameters. The self-duality extends naturally to any n umber of lattice directions: given a third discrete direction asso ciated with a shift op erator, sa y , R associated with the parameter γ , the induced sp ectral problem retains the same structure as (3.15) with co eﬃcien ts taking the same forms as h and g through { T , α } ↔ { R, γ } . 3.3 BSQ equations and deformation with resp ect to lattice parameter In tegrable BSQ-type equations arise as compatibility conditions of linear sp ectral prob- lems. The time deformations could in volv e inﬁnitely many contin uous “time v ariables” corresp onding to hierarch y of BSQ equations (see, for instance, [10]). The shift deforma- tions are induced by Darb oux transformations, and, in particular, the discrete Darb oux transformations will giv e rise to discrete BSQ equations consisten t on a multi-dimensional lattice. W e also consider deformation of the discrete sp ectral problem with resp ect to the lattice parameter. This yields a non-autonomous semi-discrete BSQ system whic h enco des the complete hierarc hy of semi-discrete BSQ equations [62, 63] Con tin uous BSQ equation: Consider the Lax pair (see, for instance, [10]) φ ′′′ + uφ ′ + v φ = λφ , ˙ φ = φ ′′ + 2 3 uφ , (3.30) where ˙ denotes deriv ativ e with resp ect to a time v ariable. Compatibilit y yields ˙ v − v ′′ + 2 3 uu ′ + 2 3 u ′′′ = 0 , ˙ u + u ′′ − 2 v ′ = 0 . (3.31) Eliminating v and their deriv ativ es, one gets the standard BSQ equation 3 ¨ u + 4( uu ′ + u ′′′ ) ′ = 0 . (3.32) 17 Semi-discrete BSQ equation: Compatibility of the following system φ ′′′ + uφ ′ + v φ = λφ , φ = φ ′ − r φ , (3.33) with ¯ shift induced by the Darb oux transformation (3.5) yields u = u + 3 r ′ , v = v + u ′ + 3 r r ′ + 3 r ′′ , v + ur + r ′′ + r 3 + 3 r r ′ + α = 0 . (3.34) By introducing a p oten tial v ariable w suc h that w − w = r , 3 w ′ = u , (3.35) the ab o ve system can b e reduced to a semi-discrete equation ( w + w + w ) ′′ = ( w − w ) 3 + ( w − w ) 3 + 3( w − w ) w ′ + 3( w − w ) w ′ = 0 , (3.36) kno wn as a semi-discrete p otential BSQ equation [59, 64]. Note that the ab ov e w is the same w as in (3.27). Discrete BSQ equation: T aking the compatibilit y of φ + h φ + g φ + αφ = λφ , e φ = φ − f φ (3.37) yields e h = h + f − f , e g = g + f h − f h + f ( f + f ) , g = β − α f − ( f + h ) f . (3.38) In terms of the p oten tial v ariable w , the ab o ve system leads to the lattice p oten tial BSQ equation deﬁned on a nine-p oint stencil [42] α − β w − e w − α − β e w − e e w = ( e w − e e w ) e e w − ( e w − w ) w + ( e e w − w ) e w . (3.39) The multi-dimensional consistency of (3.39) could b e understo od as a result of the self- dual form ulation of discrete sp ectral problems. Note that, recently , the m ulti-dimensional consistency prop ert y of the nine-p oin t lattice p oten tial BSQ equation (3.39) was explicitly v eriﬁed on a 3 × 3 × 3-vertex cubic lattice comprising 27 lattice vertices and 8 elementary cub es [61]. Non-autonomous semi-discrete BSQ system: T aking the compatibilit y of φ + h φ + g φ + sφ = 0 , φ s = − w s φ , (3.40) where · denotes the backw ard shift and · s denotes partial deriv ative with resp ect to s . Comparing to (3.13), the parameter s is set as s = α − λ with the sp ectral parameter λ b eing ﬁxed, so that ∂ s = ∂ α . (3.41) Therefore, φ s means deformation of φ with resp ect to the lattice parameter α . The compatibilit y yields the following set of semi-discrete BSQ equations g s = h w s − h w s , w s = n g . (3.42) 18 Using (3.27) as in the self-dual form ulation of the spectral problem (3.13) and eliminating g in the ab o ve system yields a non-autonomous semi-discrete equation in w only: ( n + 1) w ss w 2 s =  w − w  w s −  w − w  w s , (3.43) whic h w as kno wn as the semi-discr ete gener ating e quation for the (p oten tial) BSQ hier- arc hy in the sense that it enco des the complete hierarc hy of semi-discrete BSQ equations (including (3.36) and higher-order equations) [62, 63]. The pair (3.40) will play a crucial role in later deriv ation of the PGL(3)-in v arian t generating systems (see Section 4.3). Remark 3.5 Soliton-typ e solutions for the ab ove BSQ e quations c an b e expr esse d in close d form via the Darb oux–Crum formulae that ar e given in App endix C. 4 PGL(3) -in v ariant BSQ systems and generating systems This section ﬁrst presents the contin uous and discrete PGL(3)-inv ariant BSQ systems (see (4.2) and (4.13) resp ectiv ely) in terms of the common dep enden t v ariables z 1 , z 2 that are inhomogeneous co ordinates of the Lax pairs of BSQ equations. These systems are natural rank-3 analogues of contin uous and discrete Sch w arzian KdV equations. A remark able feature is that the PGL(2)-in v arian t (Sch w arzian) BSQ equations—b oth con tinuous and discrete—can b e recov ered from their PGL(3) counterparts via a lifting-de c oupling me ch- anism . By considering deformations with resp ect to the lattice parameters, we derive tw o additional PGL(3)-in v arian t systems in z 1 , z 2 : a non-autonomous semi-discrete system (4.38) and a coupled system of fourth-order PDEs with the lattice parameters serving as indep enden t v ariables (4.67). W e refer to them as PGL(3)-in v arian t generating systems as they enco de the complete hierarchies of semi-discrete and con tinuous BSQ equations, resp ectiv ely . 4.1 Con tin uous PGL(3) -in v arian t BSQ system W e b egin b y deriving the con tinuous PGL(3)-in v arian t BSQ system from the Lax pair (3.30). Let φ 1 , φ 2 , φ b e a basis of solutions to the sp ectral problem. In terms of the inhomogeneous co ordinates z 1 = φ 1 /φ, z 2 = φ 2 /φ , (4.1) the compatibility of Lax pair yields a coupled system: ˙ z 1 − z ′′ 1 z ′ 1 = ˙ z 2 − z ′′ 2 z ′ 2 = − 2 3 z ′ 1 z ′′′ 2 − z ′ 2 z ′′′ 1 z ′ 1 z ′′ 2 − z ′ 2 z ′′ 1 . (4.2) This system possesses the remark able prop ert y of being inv ariant under the PGL(3) action (2.10). This can b e veriﬁed by comparing with the generating inv ariants R 1 , R 2 in (2.52): the system (4.2) is equiv alen t to the conditions R 1 = 1 and R 2 = 0, which are manifestly PGL(3)-in v arian t. Therefore, w e refer to (4.2) as the PGL(3) -invariant BSQ system . The deriv ation of (4.2) is straightforw ard. Insert φ 1 = z 1 φ , φ 2 = z 2 φ in to (3.30). It follows from the third-order linear sp ectral problem (the ﬁrst equation in (3.30)) that (2.12) holds. The second equation in (3.30) (the time ev olution part of the Lax pair) yields ˙ z i = z ′′ i + 2( φ ′ /φ ) z i , i = 1 , 2 . (4.3) Com bining (2.12) and (4.3) leads to (4.2). 19 There is a well-kno wn one-to-one corresp ondence b etw een the third-order linear dif- feren tial op erator and classes of pro jectiv ely equiv alen t nondegenerate curves in P 2 [50]. Viewing z 1 , z 2 as an aﬃne chart of a nondegenerate pro jectiv e curve in P 2 (nondegeneracy means z ′ 1 z ′′ 2 − z ′ 2 z ′′ 1  = 0), the lift to ( φ 1 , φ 2 , φ ) corresp onds precisely to the Lax pair of the BSQ equation (3.30). Therefore, system (4.2) go v erns a class of pro jectiv ely equiv alent nondegenerate curves in P 2 ev olving along the BSQ ﬂow. It represen ts a natural rank-3 analogue of the Sch w arzian KdV equation. Remark 4.1 A closely r elate d system to (4.2) was derive d in [49] when c onsidering c on- tinuum limit of p en tagram maps on p olygons in P 2 . In fact, (4.2) is e quivalent to the formula in R emark 6 . 6 of [49] by the lift of z 1 , z 2 to Γ = ( z 1 , z 2 , 1) . Another remark able prop ert y of the PGL(3)-in v arian t BSQ system (4.2) is that each comp onen t independently satisﬁes a PGL(2)-inv ariant equation, whic h is kno wn as the Schwarzian BSQ e quation [66]. Prop osition 4.2 L et z 1 , z 2 b e smo oth solutions of the PGL(3) -invariant BSQ system (4.2) . Then, e ach c omp onent indep endently satisﬁes the PGL(2) -invariant e quation 3 ∂ t  ˙ z z ′  + ∂ x S [ z ] + 3 2  ˙ z z ′  2 ! = 0 , (4.4) known as the Schwarzian BSQ e quation [66], wher e S [ z ] denotes the Schwarzian derivative. Pro of: The pro of relies on lifting the system (4.2) to a three-comp onen t system via an auxiliary v ariable ξ = z ′ 2 z ′ 1 . (4.5) One can sho w that the resulting three-comp onen t system inv olving ( z 1 , z 2 , ξ ) decouples in to t wo t wo-component systems inv olving ( z 1 , ξ ) and ( z 2 , ξ ), resp ectiv ely . Eac h of these can b e further reduced to a single-v ariable equation. A natural geometric in terpretation of this mec hanism is presen ted in Remark 4.3 b elo w. Precisely , inserting (4.5) into (4.2) yields a three-comp onen t system in ( z 1 , z 2 , ξ ): z ′ 2 z ′ 1 = ˙ z 2 − z ′′ 2 ˙ z 1 − z ′′ 1 = ξ , ˙ z 1 z ′ 1 = − 2 3 ξ ′′ ξ ′ − 1 3 z ′′ 1 z ′ 1 . (4.6) Diﬀeren tiating the ﬁrst equation in (4.6) with resp ect to x allo ws us to eliminate z 2 : ˙ ξ = ξ ′′ +  ˙ z 1 + z ′′ 1 z ′ 1  ξ ′ , (4.7) whic h together with the second equation in (4.6) forms a system in ( z 1 , ξ ). One can express this system as ˙ z 1 z ′ 1 = − ˙ ξ + ξ ′′ 2 ξ ′ , z ′′ 1 z ′ 1 = 3 ˙ ξ − ξ ′′ 2 ξ ′ . (4.8) Eliminating ξ or z 1 yields a single-v ariable equation in z 1 or ξ . Eac h of the equations is precisely the Sc hw arzian BSQ equation (4.4). Similar pro cedures can b e applied to z 2 . 20 Remark 4.3 (Geometric interpretation.) The lifting and decoupling me chanism ad- mits the fol lowing ge ometric pictur e. Consider the lift of z 1 , z 2 to a curve Γ = ( z 1 , z 2 , 1) , then the lifte d dual curve Ξ is given by Ξ = Γ × Γ ′ =  − z ′ 2 , z ′ 1 , z ′ 1 z 2 − z ′ 2 z 1  , (4.9) which, by deﬁnition, is incident to Γ . The auxiliary variable ξ c an b e identiﬁe d with one of the aﬃne c o or dinates of the dual pr oje ctive curve, and the triple ( z 1 , z 2 ; ξ ) p ar ametrises an inciden t manifold with a vanishing form dz 2 − ξ dz 1 = 0 . This form is pr eserve d by the BSQ ﬂow, i.e. (4.6) , and al lows elimination of al l derivatives of z 2 in favour of z 1 and ξ . No w let us see the natural group of transformations on the t wo-component system (4.8). Let ξ 1 , ξ 2 b e an aﬃne c hart of the dual curv e Ξ with ξ 1 = ξ , ξ 2 = z ′ 2 z 1 − z ′ 1 z 2 z ′ 1 = z 1 ξ 1 − z 2 . (4.10) A PGL(3) action (2.10) induces a PGL(3) transformation on ξ 1 , ξ 2 . Consider a standard em b edding of PGL(2) into PGL(3) acting on z 1 , z 2 as ( z 1 , z 2 ) 7→  m 11 z 1 + m 31 m 13 z 1 + m 33 , z 2 m 13 z 1 + m 33  , (4.11) with ∆ = m 11 m 33 − m 13 m 31  = 0. This yields ( ξ 1 , ξ 2 ) 7→ ( m 33 ξ 1 + m 13 ξ 2 , m 31 ξ 1 + m 11 ξ 2 ) . (4.12) One can chec k that (4.8) is inv ariant under the ab o ve PGL(2) action b y taking account of (4.10). This implies that the single-v ariable equation in z 1 , i.e. the Sch w arzian BSQ equation (4.4), is PGL(2)-inv ariant. 4.2 Discrete PGL(3) -in v arian t BSQ system W e now turn to the exact discretisation of (4.2) of the contin uous system (4.2). Recall the Lax pair of the discrete BSQ equation (3.37). T aking the inhomogeneous co ordinates z 1 , z 2 from the basis solutions φ 1 , φ 2 , φ as in (4.1), the compatibilit y condition yields a coupled system for z 1 , z 2 of the form z 1 − z 1 e z 1 − z 1 = z 2 − z 2 e z 2 − z 2 = β − λ α − β    e z , e z , z       e z , e z , e z    , z = (1 , z 1 , z 2 ) ⊺ , (4.13) deﬁned on a ﬁv e-p oin t stencil in a square lattice. Note that the equalit y of the ﬁrst t wo terms in (4.13) can b e equiv alen tly written as | z , z , e z | = 0 , (4.14) deﬁned on a three-p oin t stencil. This equation yields the equality z 1 − z 1 e z 1 − z 1 = z 2 − z 2 e z 2 − z 2 =   z , z , z     z , e z , z   , (4.15) whic h allo ws us to express the second equalit y of (4.13) as α − β β − λ =    e z , e z , z       e z , e z , e z      z , e z , z     z , z , z   =    e z , e z , z       e z , e z , z       e z , e z , e z       e z , z , z    ·    e z , z , z      z , e z , z      e z , e z , z      z , z , z   . (4.16) 21 Comparing with the generating in v arian ts (2.57), (4.16) is manifestly PGL(3)-inv ariant. Therefore, w e refer to (4.13) as the discr ete PGL(3) -invariant BSQ system . It constitutes an exact discretisation of the con tinuous system (4.2) in the sense of Shabat [58, 59]. The system (4.13) can b e derived as follo ws: substituting φ 1 = z 1 φ, φ 2 = z 2 φ in to the discrete deformation part, i.e. the second equation in (3.37), yields e φ φ = φ φ − f , e z i e φ φ = z i φ φ − z i f , i = 1 , 2 . (4.17) Arranging the abov e equations leads to the following expressions for f f = e z 1 − z 1 e z 1 − z 1 φ φ = e z 2 − z 2 e z 2 − z 2 φ φ = e z 1 − z 1 z 1 − z 1 e φ φ = e z 2 − z 2 z 2 − z 2 e φ φ . (4.18) Also insert φ 1 = z 1 φ, φ 2 = z 2 φ in to the ﬁrst equation in (3.37) whic h is the linear diﬀerence equation studied in Section 3.2. It follo ws from the factorization (3.15) that ( λ − β ) φ = e φ − ( f 3 + f 2 ) e φ + f 2 f 3 e φ , (4.19a) ( λ − β ) z i φ = e z i e φ − ( f 3 + f 2 ) e z i e φ + f 2 f 3 e z i e φ , i = 1 , 2 , (4.19b) with f 2 f 3 = ( β − α ) /f . This allows us to obtain 1 f e φ φ = β − λ α − β    e z , e z , z       e z , e z , e z    . (4.20) Com bining (4.18) and (4.20) leads to (4.13). Bey ond its PGL(3)-inv ariance, the system (4.13) p ossesses t w o remark able features: ﬁrst, it can b e “lifted” to a three-comp onen t quad-system that is m ulti-dimensional consis- ten t; second, eac h of its comp onen ts satisﬁes independently the discrete PGL(2)-in v arian t BSQ equation kno wn as the Sch w arzian BSQ equation [66]. The lift of (4.13) can b e done by in tro ducing an auxiliary ﬁeld y y = e z 2 − z 2 e z 1 − z 1 = z 2 − z 2 z 1 − z 1 . (4.21a) This allows us to get a quad-equation in volving z 1 , y in the form e y − y e y − e y = ( α − λ ) ( β − λ ) ( z 1 − z 1 ) ( e z 1 − z 1 ) ( e z 1 − e z 1 ) ( e z 1 − z 1 ) . (4.21b) W e regard (4.21) as a three-comp onent system. Alternativ e forms of (4.21a) ( e z 1 − z 1 ) y = e z 2 − z 2 , ( z 1 − z 1 ) y = z 2 − z 2 , (4.22) can b e seen as “side” equations deﬁned on edges of a quadrilateral. T aking resp ectively ¯ and e shifts of the ﬁrst and second equations in (4.22) yields expressions of e z 1 , e z 2 : e z 1 = y z 1 − e y e z 1 + e z 2 − z 2 y − e y , e z 2 = y e y ( z 1 − e z 1 ) + y e z 2 − e y z 2 y − e y . (4.23) T ogether with (4.21b), one could express e y in terms of z 1 , z 2 , y and their ﬁrst-order shifts. This allo ws us to set up an initial-v alue problem on an elementary quadrilateral, and also on an elemen tary cub e in a multi-dimensional lattice. 22 Theorem 4.4 (Multi-dimensional consistency .) The thr e e-c omp onent quad-system (4.21) is c onsistent ar ound an elementary cub e in a multi-dimensional lattic e. Rev ersely , b y inserting the expressions of e y obtained resp ectively by shifts of (4.22) in to (4.21b), one could recov er (4.13), and its natural companion PGL(3)-inv ariant system e z 1 − z 1 z 1 − e z 1 = e z 2 − z 2 z 2 − e z 2 = α − λ β − α    e e z , e z , z       e e z , e z , z    , (4.24) whic h can also be derived from (4.13) b y interc hanging ( e , α ) and ( ¯ , β ). Therefore, the three-comp onen t quad-system (4.21) “cov ers” b oth (4.13) and (4.24). Similar to the con tin uous case, each comp onent of (4.13) indep enden tly satisﬁes a nine-p oin t PGL(2)-inv ariant equation known as the lattice Sch w arzian BSQ equation [39]. Prop osition 4.5 Each c omp onent z 1 , z 2 of the discr ete PGL(3) -invariant BSQ system (4.13) indep endently satisﬁes the discr ete PGL(2) -invariant e quation ( α − λ )( e e z − e e z )( e z − e z ) − ( β − λ )( e e z − e z )( e e z − e z ) ( β − λ )( e z − z )( e z − z ) − ( α − λ )( e z − e z )( z − z ) = ( e e z − e e z )( e z − e e z )( e z − z ) ( e e z − e z )( e z − z )( z − z ) , (4.25) known as the lattic e Schwarzian BSQ e quation [39]. Pro of: Using (4.22) and the ﬁrst equation in (4.23), one could eliminate z 2 and get a t wo-component system in ( z 1 , y ): e z 1 = ( y − y ) z 1 − ( e y − y ) e z 1 y − e y , e y − y e y − e y = ( α − λ ) ( β − λ ) ( z 1 − z 1 ) ( e z 1 − z 1 ) ( e z 1 − e z 1 ) ( e z 1 − z 1 ) . (4.26) F or simplicit y , let A = y − y , B = e y − y , then e A −B = A − B . Also let P = z 1 − z 1 , Q = e z 1 − z 1 and C = P e P QQ , then (4.26) can b e written as A Q = B e P , B e A = α − λ β − λ C , (4.27) and one can also get A = ( B − e A ) e P Q − e P , B = ( B − e A ) Q Q − e P , A B = e P Q . (4.28) T aking bac kward shifts of (4.27): B e = α − λ β − λ A C e , e A = β − λ α − λ B C , α − λ β − λ C e = B e A = ( B e − A ) ( B − e A ) Q e ( Q − e P ) e P ( Q e − P ) , (4.29) yields B e − A B − e A = A ( α − λ ) C (( α − λ ) C e − ( β − λ )) B ( β − λ )(( α − λ ) C − ( β − λ )) . (4.30) Using the ab o ve relations leads to an equation inv olving P , Q , C only which is precisely a bac kward-shifted cop y of (4.25). Similarly , one could eliminate z 1 whic h leads to the same nine-p oin t single-v ariable equation in y . F rom (4.21), one can also obtain a t wo-component system in z 2 , y which can b e decoupled to a single-v ariable equation in z 2 . 23 Remark 4.6 T o the b est of our know le dge, the thr e e-c omp onent quad-e quation (4.21) , as wel l as its “pr oje ctions” (4.13) and (4.24) , wer e ﬁrst identiﬁe d in [61] as r e ductions of a Q3 analo gue of the lattic e BSQ e quation. It app e ar e d as a natur al r ank- 3 extension of the cr oss-r atio e quation (also known as the discr ete Schwarzian KdV e quation, or Q1 ( δ = 0) in the A d ler–Bob enko–Suris classiﬁc ation [2]), and r epr esente d a genuine extension of discr ete inte gr able system of BSQ typ e gener alizing pr evious known examples [17, 18, 42, 62, 63, 72]. While the system (4.21) was ﬁrst derive d in [61] thr ough r e ductions, our curr ent appr o ach demonstr ates it emer ges intrinsic al ly fr om the factorise d sp e ctr al pr oblem (3.37) natur al ly asso ciate d with the PGL(3) symmetry. The discrete PGL(3)-in v arian t system (4.13) admits a similar geometric in terpretation analogous to its con tinuous counterpart (4.2). The inhomogeneous co ordinates z 1 , z 2 whic h arise from the third-order spectral problem, characterize a class of pro jectiv ely equiv alen t nondegenerate discr ete pr oje ctive curves (or p olygons) in P 2 [49]. The lift to ( φ 1 , φ 2 , φ ) ev olves precisely according to the discrete Lax pair (3.37). Due to the cov ariance of the discrete sp ectral problem (as discussed in Section 3.2), the companion system (4.24) admits the same in terpretation simply b y in terchanging the lattice directions. A dual discr ete curve Ξ to Γ = ( z 1 , z 2 , 1) can b e obtained through the following incident relation Ξ = Γ × Γ = ( z 2 − z 2 , z 1 − z 1 , z 2 z 1 − z 1 z 2 ) , (4.31) whic h makes the auxiliary v ariable y given by (4.21) as one of the inhomogeneous co or- dinates of Ξ. This implies that ( z 1 , z 2 ; y ) are living in a “discrete inciden t space” on P 2 , and the inciden t relation (4.31) is preserv ed by the discrete dynamics (4.21). Let y 1 = y , y 2 = z 1 z 2 − z 2 z 1 z 1 − z 1 = z 2 − z 1 y 1 (4.32) b e the inhomogeneous coordinates of Ξ. One can show that under the PGL(2) action (4.11), ( y 1 , y 2 ) transform exactly as (4.12), and the tw o-comp onen t system (4.26) is in- v arian t. Therefore, (4.26) p ossesses a natural PGL(2) symmetry implying the PGL(2) symmetry of the lattice Sch w arzian BSQ equation (4.25). Remark 4.7 Using the L ax p airs of semi-discr ete BSQ e quations, one c ould derive semi- discr ete PGL(3) -invariant BSQ systems in an analo gous manner. F or instanc e, (3.33) le ads to z ′ 1 z ′′′ 2 − z ′ 2 z ′′′ 1 z ′ 1 z ′′ 2 − z ′ 2 z ′′ 1 − z ′ 1 z ′′′ 2 − z ′ 2 z ′′′ 1 z ′ 1 z ′′ 2 − z ′ 2 z ′′ 1 = 3 (log ζ ) ′ , ζ = z ′ 1 z 1 − z 1 = z ′ 2 z 2 − z 2 . (4.33) 4.3 PGL(3) -in v ariant generating systems The notion of a gener ating PDE for the KdV hierarch y w as ﬁrst introduced in [44]. It ap- p eared as a co v arian t (with respect to the indep enden t v ariables) scalar integrable PGL(2)- in v arian t PDE in which the lattice parameters of the corresp onding lattice KdV equation serv e as independent v ariables, while the discrete lattice v ariables play the role of parame- ters. Through a systematic expansion, this generating PDE yields the complete hierarch y of KdV equations. A Schwarzian KdV (or PGL(2) -KdV) gener ating PDE , serving as a “pro jective” formulation in terms of the inhomogeneous v ariable w as also developed in [44], enco ding the complete hierarc hy of Sch w arzian KdV equations. In [62, 63], an analogous system of generating PDEs w as presen ted for the BSQ hierarc hy together with its La- grangian structure. This coupled system is PGL(3)-in v arian t and is asso ciated with the complete hierarch y of BSQ equations. 24 The generating PDEs p ossess signiﬁcant ph ysical relev ance: the generating PDE for the KdV hierarc h y is a generalisation of the Ernst equation of General Relativity describing gra vitational wa v es, while the one for the BSQ hierarch y coincides with the Ernst equa- tions arising in the Einstein–Maxwell–W eyl theory , which describ es gravitational wa v es in the presence of neutrino ﬁelds sub ject to electromagnetic interaction. This remark able connection betw een w ater w av e theory (KdV, BSQ) and gra vitational w a ve theory remains to b e fully understoo d. In this subsection, we deriv e the pro jective form ulation (in terms of z 1 , z 2 ) of the BSQ generating PDEs. The deriv ation relies on deformations of the discrete sp ectral problems with resp ect to the lattice parameters, c.f. (3.40). W e b egin by deriving a non-autonomous semi-discrete system (4.38) in Section 4.3.1, referred to as the semi- discr ete PGL(3) -invariant gener ating system , playing a role analogous to the semi-discrete BSQ generating equation (3.43). Then in Section 4.3.2, we present the PGL(3)-inv ariant generating PDE, ﬁrst as a coupled system of PDEs for determinants, and subsequen tly in vectorial form with its Lagrangian structure. The generating PDE corresp onds to the en tire BSQ hierarch y . While the explicit forms of the PGL(3)-inv ariant generating systems are established here, throughout analysis of their further properties is b ey ond the scope of the present w ork and will b e reserv ed for future in vestigations. These includes the hierarch y-generating mec hanism; possible geometric in terpretations; symmetry analysis and reductions to higher- rank P ainlev´ e-t yp e equations; explicit connections to ph ysical models such as the Einstein– Maxw ell–W eyl theory . 4.3.1 A semi-discrete PGL(3) -inv ariant generating system Let φ 1 , φ 2 , φ b e a basis of solutions to (3.40). Inserting z i φ = φ i ( i = 1 , 2) into the deformation part φ s = − w s φ yields z s φ + z φ s = − w s z φ = z φ s ⇒ ∂ s log φ = ∂ s z 1 z 1 − z 1 = ∂ s z 2 z 2 − z 2 , (4.34) where · denotes the bac kward shift in the lattice v ariable n , and z denotes the three- comp onen t vector z = (1 , z 1 , z 2 ) ⊺ . T aking a bac kward shift of the sp ectral problem, one obtains z φ + h z φ + g z φ + s z φ = 0 , (4.35) whic h implies g   z , z , z   φ + s   z , z , z   φ = 0 . (4.36) Using again φ s = − w s φ and taking w s = n g in (3.42), one obtains ∂ s log φ = n s   z , z , z     z , z , z   . (4.37) Com bining (4.34) and (4.37) leads to ∂ s z 1 z 1 − z 1 = ∂ s z 2 z 2 − z 2 = n s   z , z , z     z , z , z   , (4.38) whic h is a PGL(3)-inv ariant non-autonomous diﬀerential-diﬀerence system in z 1 , z 2 with s, n b eing the indep enden t v ariables. Similarly , one can obtain a cov ariant form ulation of (3.40) using the dual sp ectral problem (3.25): e e e φ + p e e φ + q e φ + tφ = 0 , φ t = − w t φ , (4.39) 25 with t = β − λ . This leads to a dual system ∂ t z 1 z e 1 − z 1 = ∂ t z 2 z e 2 − z 2 = m t   e e z , e z , z     e e z , e z , z e   , (4.40) with indep enden t v ariables t, m . Remark 4.8 In addition to the non-autonomous diﬀer ential-diﬀer enc e systems (4.38) and (4.40) , we have also the c omp atible autonomous diﬀer ential-diﬀer enc e systems ∂ τ z 1 z 1 − z 1 = ∂ τ z 2 z 2 − z 2 = α ′ α − λ   z , z , z     z , z , z   , (4.41a) ∂ σ z 1 z 1 − z e 1 = ∂ σ z 2 z 2 − z e 2 = β ′ β − λ   e e z , e z , z     e e z , e z , z e   , (4.41b) in which α ′ = 3 α 2 / 3 , β ′ = 3 β 2 / 3 , and wher e τ and σ ar e Miw a v ariables whose ve ctor ﬁelds c ontain the diﬀer ential ve ctor ﬁelds of the entir e hier ar chy of ﬂows, i.e. ∂ τ = ∞ X j =0 α − ( j +1) / 3 ∂ t j , ∂ σ = ∞ X j =0 β − ( j +1) / 3 ∂ t j , (4.42) wher e the t j ar e the higher time-variables of the BSQ hier ar chy. The e quations (4.41) ar e derive d in a similar way as the non-autonomous system, namely by e quipping the thir d- or der diﬀer enc e sp e ctr al pr oblems (3.40) and (4.39) with the time-deformation e quations ∂ τ φ = − w τ φ and ∂ σ φ = − w σ φ e . (4.43) The diﬀer enc e with the s and t -derivatives is that ther e is no explicit τ or σ -dep endenc e in the sp e ctr al pr oblems, unlike (3.40) and (4.39) which dep end explicitly on s or t . A lternatively, Eqs. (4.41) c an b e derive d by applying a “skew c ontinuum limit” (c.f. [19]) of the diﬀer enc e-diﬀer enc e system (4.13) , but we omit the details. Wher e as the ve ctor ﬁelds asso ciate d with the non-autonomous diﬀer ential-diﬀer enc e e quations c an b e r e gar de d as master symmetries of the hier ar chy of PGL(3) invariant diﬀer ential-diﬀer enc e hier ar chy, gener ate d by exp anding (4.41) in p owers of α or β , the latter form the hier ar chy of c ontinuous higher-or der symmetries of the ful ly discr ete system (4.13) , c.f. e.g. [70]. 4.3.2 A PGL(3) -in v arian t generating PDE F rom here on, the 2-comp onent v ector z = ( z 1 , z 2 ) ⊺ will b e used rather than the 3- comp onen t vector (1 , z 1 , z 2 ) ⊺ that was used in the previous subsection. In terms of their deriv ativ es, and their 2 × 2 determinants, the PGL(3)-inv arian t generating PDE associated with the previous semi-discrete and fully discrete equations can be describ ed as a coupled system inv olving tw o auxiliary scalar v ariables P and Q as follo ws. Theorem 4.9 The gener ating PDE, a c ouple d system of PDEs in terms of the variables s and t , is obtaine d fr om the fol lowing c ouple d system by eliminating the two auxiliary sc alar variables P , Q : P s =  1 − n s − | z t , z ss | | z s , z t |  P + s t | z s , z ss | | z s , z t | Q , (4.44a) Q t =  1 − m t + | z s , z tt | | z s , z t |  Q − t s | z t , z tt | | z s , z t | P , (4.44b) 26 and P t + Q s = 2 | z s , z st | | z s , z t | Q − 2 | z t , z st | | z s , z t | P . (4.44c) By eliminating the auxiliary v ariables P , Q from the system (4.44), according to the computation outlined in App endix D, w e can derive the following coupled system in terms of the 2 × 2 determinants of the deriv ativ es of the v ector z : ∂ s " 2  | z t , z st | | z s , z t |  2 +  | z s , z tt | | z s , z t | − m − 1 t  2 −  | z t , z ss | | z s , z t | + 2 | z s , z st | | z s , z t | + n − 1 s  t s | z t , z tt | | z s , z t | # − 2  | z s , z tt | | z s , z t | − m − 1 t  ∂ t  | z s , z st | | z s , z t |  − ∂ t  | z t , z ss | | z s , z t |   2 | z t , z st | | z s , z t | − | z s , z tt | | z s , z t | + m − 1 t  +2 ∂ t  s t | z s , z ss | | z s , z t |  t s | z t , z tt | | z s , z t | + s t | z s , z ss | | z s , z t | ∂ t  t s | z t , z tt | | z s , z t |  = 2 ∂ s ∂ t  | z t , z st | | z s , z t |  − ∂ 2 s  t s | z t , z tt | | z s , z t |  − ∂ 2 t  | z t , z ss | | z s , z t |  , (4.45a) ∂ t " 2  | z s , z st | | z s , z t |  2 +  | z t , z ss | | z s , z t | + n − 1 s  2 −  | z s , z tt | | z s , z t | + 2 | z t , z st | | z s , z t | − m − 1 t  s t | z s , z ss | | z s , z t | # − 2  | z t , z ss | | z s , z t | + n − 1 s  ∂ s  | z t , z st | | z s , z t |  − ∂ t  | z t , z ss | | z s , z t |   2 | z s , z st | | z s , z t | − | z t , z ss | | z s , z t | − n − 1 s  +2 ∂ s  t s | z t , z tt | | z s , z t |  s t | z s , z ss | | z s , z t | + t s | z t , z tt | | z s , z t | ∂ s  s t | z s , z ss | | z s , z t |  = − 2 ∂ s ∂ t  | z s , z st | | z s , z t |  + ∂ 2 s  | z s , z tt | | z s , z t |  + ∂ 2 t  s t | z s , z ss | | z s , z t |  . (4.45b) This is one form of the PGL(3)-inv ariant generating PDE; its vector form, which admits a Lagrangian structure, will be in tro duced later. Before pro ceeding, a detailed deriv ation of the system (4.44) is presen ted ﬁrst. The deriv ation is quite nontrivial, relying on in tricate compatibilit y conditions of the factorised discr ete sp ectral problems and the in terpla y with the underlying discrete structure of lattice shifts. Deriv ation of (4.44) : Collect the following set of linear equations for φ : φ + h φ + g φ + sφ = 0 , h = w − w , φ s = − w s φ , (4.46a) e e e φ + p e e φ + q e φ + tφ = 0 , p = e e e w − w , φ t = − w t φ e , (4.46b) and e φ + e w φ = φ + w φ , (4.46c) where the ﬁrst tw o equations are just copies of (3.40) and (4.39) with the deformation v ariables s, t related to the lattice parameters α, β as s = α − λ , t = β − λ ⇒ ∂ s = ∂ α , ∂ t = ∂ β . (4.47) Their compatibility yields ( c.f. Section 3.3) g s = h w s − h w s , w s g = n , q t = p e e w t − p e w s , w t q e = m . (4.48) Shifts of φ in diﬀerent directions are linked via (4.46c) ( c.f. Section 3.2). It follo ws from (4.46c) that (b y shifting (4.46c) and using (4.46c) for lo wer-order shifts substitutions) e e φ + e e w e φ + e w e w φ = φ + w φ + w e w φ . (4.49) 27 The ab ov e set of equations from (4.46) to (4.49) is all w e need to derive the system (4.44), using the follo wing steps: 1. Compute φ st and φ ts using the fundamen tal deformations φ s = − w s φ and φ t = − w t φ e . Com bine expressions of φ s and φ b t as bac kward shifts of the fundamental deformations, and use (4.46c) to get φ e . The compatibility yields w st = w t w e s − w s w t w − w e , (4.50a) and φ st = w t w e s w s ( w − w e ) φ s − w s w t w t ( w − w e ) φ t . (4.50b) 2. Let φ 1 , φ 2 , φ b e a basis of solutions for (4.46). Let z = ( z 1 , z 2 ) ⊺ , and insert z 1 = φ 1 /φ , z 2 = φ 2 /φ into φ s = − w s φ , φ t = − w t φ e : z s = ( z − z ) ∂ s log φ , z t = ( z e − z ) ∂ t log φ . (4.51) Insert z 1 = φ 1 /φ , z 2 = φ 2 /φ into (4.46c): ( z − z ) φ φ = ( e z − z ) e φ φ . (4.52) Insert z 1 = φ 1 /φ , z 2 = φ 2 /φ into (4.50b): z st + ( ∂ t log φ ) z s + ( ∂ s log φ ) z t = w t w e s w s ( w − w e ) z s − w s w t w t ( w − w e ) z t . (4.53) Con tracting (4.53) with z s and z t resp ectiv ely using 2 × 2 determinants yields | z s , z st | | z s , z t | = − ∂ s log φ − w s w t w t ( w − w e ) , | z t , z st | | z s , z t | = ∂ t log φ − w t w e s w s ( w − w e ) . (4.54) T aking accoun t of (4.50a), this leads to the relation w t  | z s , z st | | z s , z t | + ∂ s log φ  − w s  | z t , z st | | z s , z t | − ∂ t log φ  = w st , (4.55) from whic h (4.44c) is deriv ed b y in troducing the quan tities P , Q (which pla y the roles of auxiliary v ariables as in (4.44)), deﬁned as P ≡ w s φ 2 , Q ≡ w t φ 2 . (4.56) 3. W e pro ve the following relation w s w s w e t w t = s 2 t 2 | z s , z ss | | z t , z tt | w t w s . (4.57) Using (4.46a) and (4.46b), one gets induced expressions of φ s , φ ss (resp ectiv ely φ t , φ tt ) in terms of φ, φ, φ (resp ectiv ely φ, e φ, e e φ ). Inv ersely , this allows us to express φ, φ (resp ectiv ely e φ, e e φ ) in terms of φ, φ s , φ ss (resp ectiv ely φ, φ t , φ tt ). Combining with (4.46c) and (4.49) yields tw o linear equations in φ inv olving φ ss , φ tt , φ s , φ t , φ . Inserting z 1 = φ 1 /φ , z 2 = φ 2 /φ into these equations leads to (4.57). 28 4. W e ha ve from the double bac k-shifted discrete sp ectral problems (4.46a) and (4.46b): φ + h φ −  1 w s g + s w ss w s w 2 s  φ s = − s w s w s φ ss , (4.58a) e φ + p e e φ −  1 w t q e e + t w tt w e t w 2 t  φ t = − t w e t w t φ tt . (4.58b) Applying z 1 = φ 1 /φ , z 2 = φ 2 /φ again yields ( z − z ) φ φ + 1 w s w s  1 − n − s ∂ s log  w s φ 2  z s = − s w s w s z ss , (4.59a) ( e z − z ) e φ φ + 1 w t w e t  1 − m − t ∂ t log  w t φ 2  z t = − t w t w b t z tt . (4.59b) Subtracting these from each other and using (4.52), w e get the vectorial relation w t w e t w s w s  ∂ s log P + n − 1 s  z s − z ss  = t s  ∂ t log Q + m − 1 t  z t − z tt  , (4.60) b et w een z s , z t , z ss and z tt , with P, Q given in (4.56). 5. Final expressions: on the one hand, com bining (4.57), (4.60) and (4.56), w e ﬁnd the v ectorial equation s | z t , z tt |  Q t + m − 1 t Q  z t − Q z tt  = t | z s , z ss |  P s + n − 1 s P  z s − P z ss  . (4.61) On the other hand, making con tractions with z s and z t resp ectiv ely using 2 × 2 determinan ts leads to (4.44a) and (4.44b). Conv ersely , inserting the expressions for P s and Q t from (4.44a) and (4.44b) in to (4.61) we get a combination b et ween tw o v ectorial iden tities b et w een z s , z t , z ss , z tt and z st , sho wing that the system (4.44) is fully equiv alen t to the vectorial equation combined with (4.44c). Let us commen t on the PGL(3)-in v ariance of the equations (4.44) and (4.61). Recall z = ( z 1 , z 2 ) ⊺ . The PGL(3) transformation, c.f. (2.10), z ⊺ 7→ z ′ ⊺ , z ′ ⊺ =  m 11 z 1 + m 21 z 2 + m 31 m 13 z 1 + m 23 z 2 + m 33 , m 12 z 1 + m 22 z 2 + m 32 m 13 z 1 + m 23 z 2 + m 33  , (4.62) can b e alternatively written as ( z ⊺ , 1) 7→ ( z ′ ⊺ , 1) = ( z ⊺ , 1) 1 µ M =  m 11 z 1 + m 21 z 2 + m 31 m 13 z 1 + m 23 z 2 + m 33 , m 12 z 1 + m 22 z 2 + m 32 m 13 z 1 + m 23 z 2 + m 33 , 1  , (4.63) with µ = m 13 z 1 + m 23 z 2 + m 33 . Here and un til the end of this section, primes temp orarily denote the co ordinates after the transformation. Using (4.56) and the fact that w is PGL(3)-in v arian t, one has P 7→ P ′ = 1 µ 2 P , Q 7→ Q ′ = 1 µ 2 Q , (4.64) 29 whic h can b e easily veriﬁed by recognising that φ = φ 3 as we did in Section 2.1.2. In fact, the transformation implies the inv ariance of the ratios | z s , z ss | | z t , z s | , | z t , z tt | | z t , z s | , (4.65) while | z ′ t , z ′ s | = det( M ) µ 3 | z t , z s | , (4.66) and we ha ve the following transformation rules for the following ratios: | z ′ t , z ′ st | | z ′ t , z ′ s | = | z t , z st | | z t , z s | − ∂ t log µ , and | z ′ t , z ′ ss | | z ′ t , z ′ s | = | z t , z ss | | z t , z s | − 2 ∂ s log µ , and similarly for the ratios of determinan ts with s and t interc hanged. With these relations, the inv ariance of (4.44) is easily established. One can show the PGL(3)-inv ariance of the vectorial-relation (4.61) as w ell. This is v eriﬁed on the basis of the following transformation rules for the v ectors z s , z t , z ss and z tt follo wing (4.62) and (4.63): ( z ⊺ , 1) 7→ ( z ′ ⊺ , 1) = ( z ⊺ , 1) 1 µ M , ( z ⊺ s , 0) 7→ ( z ′ ⊺ s , 0) =  ( z ⊺ s , 0) − µ s µ ( z ⊺ , 1)  1 µ M , ( z ⊺ st , 0) 7→ ( z ′ ⊺ st , 0) =  ( z ⊺ st , 0) − µ t µ ( z ⊺ s , 0) − µ s µ ( z ⊺ t , 0) + µ  ∂ s ∂ t 1 µ  ( z ⊺ , 1)  1 µ M , ( z ⊺ ss , 0) 7→ ( z ′ ⊺ ss , 0) =  ( z ⊺ ss , 0) − 2 µ s µ ( z ⊺ s , 0) + µ  ∂ 2 s 1 µ  ( z ⊺ , 1)  1 µ M , and similar rules with s and t interc hanged. In v erifying the in v ariance of the v ector form one uses also that the quantit y µ , due to the fact that it is a linear combination of the comp onen ts z 1 and z 2 , also satisﬁes the relation (4.61) alb eit in component form. Finally , we ﬁnish b y presenting the vector form of the generating PDE. As explained at the end of App endix D, the determinan tal form of the generating PDE (4.45) can b e cast into a vectorial form. The actual computation is quite long, so w e only presen t the result, namely ∂ s  | z t , z st | | z s , z t |  1 | z s , z t |  n s 2 z t + m t 2 z s  − ( s − t ) ( | z t , z st | z s − | z s , z st | z t ) s t | z s , z t | 2  − ∂ t  | z s , z st | | z s , z t |  1 | z s , z t |  n s 2 z t + m t 2 z s  − ( s − t ) ( | z t , z st | z s − | z s , z st | z t ) s t | z s , z t | 2  = ∂ s ∂ t  1 | z s , z t |  n s 2 z t + m t 2 z s  − ( s − t ) ( | z t , z st | z s − | z s , z st | z t ) s t | z s , z t | 2  . (4.67) W e prefer to refer to this system, rather than (4.45), as the PGL(3) -invariant gener ating PDE , as this system actually p ossesses a Lagrangian form ulation. F urthermore, this PDE is a coupled system of tw o non-autonomous fourth-order scalar PDEs in terms of the comp onen ts z 1 , z 2 , with s, t , as b efore, serving as indep enden t v ariables (related to lattice parameters), while the discrete lattice v ariables n, m pla y the role of parameters. Notably , the system is inv ariant under the interc hange of the pairs ( s, n ) ↔ ( t, m ). This symmetry reﬂects the underlying self-duality of the discrete sp ectral problems discussed in Section 30 3.2. Based on the analogy with the rank-2 case [44], the generating PDEs (4.67) are exp ected to encode the complete hierarch y of PGL(3)-in v arian t BSQ equations through a systematic expansion. A detailed veriﬁcation of this hierarch y-generating mechanism will b e pursued in future inv estigations. The Lagrangian structure is given by the follo wing Lagrangian L ( z , s, t ; n, m ) = n s 2 | z t , z st | | z s , z t | + m t 2 | z s , z st | | z s , z t | − s − t s t | z s , z st | | z t , z st | | z s , z t | 2 , (4.68) from which (4.67) arises as the Euler-Lagrange equation. Their PGL(3)-in v ariance of the latter is clear from the follo wing Prop osition. Prop osition 4.10 The L agr angian (4.68) is invariant under PGL(3) tr ansformations up to nul l L agr angian terms, i.e. up to diver genc e terms; this c an b e, for instanc e, veriﬁe d by showing that, for al l the eight evolutionary inﬁnitesimal gener ators X given by (A.11) , pr (2) X ( L ) = Div F (4.69) holds for some 2 -tuple F = (F 1 ( s, t, z , z s , z t , . . . ) , F 2 ( s, t, z , z s , z t , . . . )) , wher e Div denotes the c ontinuous diver genc e with the indep endent variables s, t , and pr denotes the pr olonga- tion of ve ctor ﬁelds (se e, e.g. (A.3) ). Namely, the PGL(3) action is a diver genc e symmetry of the asso ciate d variational pr oblem and ther efor e ensur es the PGL(3) -invarianc e of the Euler–L agr ange e quations (4.67) (se e, e.g. [47]). One can prov e this statemen t through straightforw ard computations. Here, we remark t wo curious asp ects of the Lagrangian (4.68). Remark 4.11 A PGL(3) -invariant L agr angian (mo dulo nul l L agr angian terms) yielding a gener ating system for the BSQ hier ar chy was ﬁrst given in [62, 63]. R emarkably, the L agr angian (4.68) is r elate d to the one given in [62, 63] by an inversion of the indep endent variables ( s, t ) 7→ ( p, q ) =  1 s , 1 t  . (4.70) The same tr ansformation was known in the r ank- 2 KdV c ase, c onne cting the gener ating PDE for the KdV hier ar chy and its Schwarzian version [44]. This r emarkable fe atur e suggests a uniﬁe d hier ar chy-gener ating structur e for b oth inte gr able PDEs of arbitr ary r ank and their pr oje ctive formulations. The me aning of this inversion and its r ole in c onne cting inte gr able hier ar chies r emains curious, and warr ants further investigation. Remark 4.12 We c onje ctur e that the L agr angian (4.68) , like its PGL(2) analo gue, c.f. [44], p ossesses the asp e ct of a L agr angian multiform in the sense of [30], i.e. a diﬀer ential 2 -form in a sp ac e of many variables of the typ e s, t whose c omp onents ar e given by this L agr angian, and which is close d on solutions of the Euler–L agr ange e quations. 5 Concluding remarks Based on third-order linear sp ectral problems and their factorised forms, this paper estab- lished a uniﬁed framew ork for pro jective form ulations of PGL(3) diﬀeren tial and diﬀerence in v arian ts, as w ell as PGL(3)-inv ariant BSQ-type equations, in terms of a common set of dep enden t v ariables z 1 , z 2 that are pro jective v ariables of the sp ectral problems. This framew ork suggests a rich interpla y b et ween in v arian t theory , classical pro jectiv e geome- try , and integrable systems. Several promising directions for future inv estigation emerge from this w ork. 31 • Extending 3D-consistent systems: the three-comp onen t quad-system (4.21) con- structed in Section 4.2 represents a natural rank-3 extension of the cross-ratio equa- tion. This provides a concrete example for extending kno wn examples of m ulti- comp onen t 3D-consistent systems [17, 42, 62, 63, 72] b ey ond scalar quad-equations [2, 6, 38, 43]. F urther in vestigation may rev eal new families of BSQ-type lattice equa- tions within this framework. • Sev eral asp ects of the PGL(3)-inv ariant BSQ systems deriv ed in Sections 4.1-4.2 need to b e clariﬁed, such as their Lax pairs and explicit m ulti-soliton solutions. In particular, following [36, 69], one could consider the “Miura-to wer” asso ciated with PGL(3) within the P oisson–Lie theory , where one exp ects a ric her structure with extra inv ariant subﬁelds compared to the rank-2 case. This could also leads to Hamiltonian structures for the PGL(3)-inv ariant BSQ systems in b oth contin uous and discrete cases. • Similar to how the Sch w arzian KdV equation reduces to Painlev ´ e-type equations [39, 43, 44], it is of in terest to inv estigate similarit y reductions of the PGL(3)-in v arian t BSQ systems and their generating PDEs. Such reductions are expected to yield higher-rank P ainlev ´ e systems, suc h as Garnier-type equations, establishing a con- crete link b etw een the pro jectiv e BSQ hierarch y and the theory of isomono dromic deformations for higher-order linear problems. • The framew ork can b e readily extended to arbitrary rank N , yielding PGL( N )- in v arian t integrable systems asso ciated with N th-order sp ectral problems. F ur- ther exploration may extend the framew ork to other geometric settings asso ciated with systems of linear problems, where the underlying geometry is related to ma- trix Sc hw arzian deriv atives and “Grassmannian curves” [48, 54, 55]. Suc h extensions w ould pro vide new examples of in tegrable models b ey ond the pro jective case. • While we ha ve pro vided the explicit forms of the PGL(3)-inv ariant generating PDEs in Section 4.3, several asp ects of this system remain to be understo o d. F or ex- ample, their connection to the Einstein–Maxwell–W eyl theory [62, 63], geometric in terpretations (which extend b ey ond the pro jectiv e setting), the curious parameter in version prop erties discussed in Remark 4.11, and p ossible Lagrangian multiform structures [30] (see Remark 4.12). Ac kno wledgments FN is grateful for the frequent hospitality of the Department of Mathematics of Shanghai Univ ersity since 2017 when the pro ject w as initiated, and was partially supp orted in this p eriod by the National F oreign Exp ert Program of China (No. G2022172028L). LP was partially supp orted b y JSPS KAKENHI (JP24K06852), JST CREST (JPMJCR24Q5), and Keio Universit y (Academic Dev elopment F und, F ukuza wa F und). CZ was supp orted b y NSF C (No. 12171306). DZ was supp orted b y NSF C (Nos. 12271334 and 12411540016). App endix A The generating PGL(2) - and PGL(3) -in v arian ts In this app endix, we show how b oth the diﬀeren tial and diﬀerence generating inv ariants of PGL(2) and PGL(3) actions that shown in Section 2 can b e computed using their corresp onding inﬁnitesimal generators. 32 Notation conv en tion: In Appendices A and B, primes denote the coordinates after transformations, rather than deriv ativ es. A.1 The diﬀeren tial case Recall the PGL(2)-action ab out ( x, z ) deﬁned in (2.6): ( x, z ) 7→ ( x, z ′ ) , where z ′ = m 11 z + m 21 m 12 z + m 22 with det M  = 0 . (A.1) It is w ell kno wn that the PGL( N ) group is linearisable and the corresp onding Lie algebra is sl ( N ). Consequently , w e only ha ve to focus on SL( N )- and PSL( N )-actions for ﬁnding the corresp onding diﬀerential inv ariants. Namely , in computing the inﬁnitesimal generators, w e assume det M = 1. The corresp onding inﬁnitesimal generators are X i = η i ( x, z ) ∂ z ( i = 1 , 2 , 3) with the co eﬃcien ts η i ( x, z ) giv en b y η 1 ( x, z ) := d d m 11    M =id z ′ = 2 z , η 2 ( x, z ) := d d m 12    M =id z ′ = − z 2 , η 3 ( x, z ) := d d m 21    M =id z ′ = 1 . (A.2) Note that m 22 is replaced by a function of m 11 , m 12 and m 21 around M = id using the condition det M = 1. Their prolongations to the k th-order pr ( k ) X i are (see, e.g. [47]) pr ( k ) X i = η i ( x, z ) ∂ z + ( D x η i ( x, z )) ∂ z x + · · · +  D k x η i ( x, z )  ∂ z ( k ) . (A.3) A function I ( x, z , z x , . . . ) is a diﬀerential inv arian t in volving up to k th-order deriv atives of z for some p ositiv e in teger k if and only if it satisﬁes the inﬁnitesimal in v ariance condition ( c.f. [46]) that pr ( k ) X i ( I ) = 0 (A.4) holds for every inﬁnitesimal generator X i . This yields the following system of linear PDEs     0 2 z 2 z x . . . 2 z ( k ) 0 − z 2 − 2 z z x . . . − ( z 2 ) ( k ) 0 1 0 . . . 0            I x I z I z x . . . I z ( k )        =   0 0 0   . (A.5) Let k = 3; using Gaussian elimination and prop erly arranging ro ws of the co eﬃcien t matrix, the abov e system b ecomes     0 1 0 0 0 0 0 z x 0 z xxx − 3 z 2 xx z x 0 0 0 z x 3 z xx           I x I z I z x I z xx I z xxx       =   0 0 0   . (A.6) This linear system of PDEs can b e solv ed b y the characteristic metho d. The ﬁrst row giv es I = I ( x, z x , z xx , z xxx ); substituting it to the last ro w, i.e. z x I z xx + 3 z xx I z xxx = 0, w e obtain I = I  x, z x , z xxx − 3 z 2 xx 2 z x  . (A.7) 33 This is then substituted to the ﬁnal equation z x I z x +  z xxx − 3 z 2 xx z x  I z xxx = 0, and w e get I = I  x, z xxx z x − 3 z 2 xx 2 z 2 x  . (A.8) This amounts to the generating diﬀeren tial in v arian ts x and the Sc hw arzian deriv ativ e S [ z ] = z xxx z x − 3 z 2 xx 2 z 2 x . (A.9) There exist no other diﬀerential in v arian ts with order lo wer than the Sch warzian deriv ativ e; all higher-order diﬀerential inv ariants are functions of the generating inv ariants x , the Sc hw arzian deriv ativ e S [ z ] and inv ariant deriv ativ es of S [ z ] with resp ect to x . Generating diﬀeren tial inv ariants with resp ect to the PGL(3)-action deﬁned in (2.10), namely      z ′ 1 = m 11 z 1 + m 21 z 2 + m 31 m 13 z 1 + m 23 z 2 + m 33 , z ′ 2 = m 12 z 1 + m 22 z 2 + m 32 m 13 z 1 + m 23 z 2 + m 33 , (A.10) can b e calculated using the same metho d. They are the S 1 [ z 1 , z 2 ] and S 2 [ z 1 , z 2 ] giv en by (2.14) and x , which form the set of generating diﬀeren tial inv ariants. W e omit the details here. Note that the eight inﬁnitesimal generators are 2 z 1 ∂ z 1 + z 2 ∂ z 2 , z 1 ∂ z 2 , − z 2 1 ∂ z 1 − z 1 z 2 ∂ z 2 , z 2 ∂ z 1 , z 1 ∂ z 1 + 2 z 2 ∂ z 2 , − z 1 z 2 ∂ z 1 − z 2 2 ∂ z 2 , ∂ z 1 , ∂ z 2 , (A.11) corresp onding to the parameters m 11 , m 12 , m 13 , m 21 , m 22 , m 23 , m 31 and m 32 , resp ectiv ely . A.2 The diﬀerence case Similarly as in App endix A.1, a function is a diﬀerence inv ariant if and only if it satisﬁes the discrete v ersion of inﬁnitesimal in v ariance condition, whic h is prop osed b elow. In the discrete case, prolongations of inﬁnitesimal generators X i = η i ( n, z ) ∂ z are giv en b y (see, e.g. [23]) pr X i = · · · + η i ( n, z ) ∂ z + η i ( n + 1 , z ) ∂ z + η i  n + 2 , z  ∂ z + η i  n + 3 , z  ∂ z + · · · , (A.12) and recall η 1 ( n, z ) = 2 z , η 2 ( n, z ) = − z 2 , and η 3 ( n, z ) = 1 for the PGL(2)-action (see (A.2)). A diﬀerence inv ariant I = I  n, z , z , z , z  satisﬁes the inﬁnitesimal in v ariance condition pr X i ( I ) = 0 for ev ery X i , which leads to a system of linear PDEs      2 z 2 z 2 z 2 z − z 2 − z 2 −  z  2 −  z  2 1 1 1 1          I z I z I z I z     =   0 0 0   . (A.13) 34 This can b e solved similarly using Gaussian elimination and c haracteristic metho d for linear PDEs. W e ﬁrst obtain an equiv alen t system           1 0 0  z − z  z − z  ( z − z ) ( z − z ) 0 1 0 −  z − z  z − z  ( z − z ) ( z − z ) 0 0 1  z − z  z − z  ( z − z )( z − z )               I z I z I z I z     =   0 0 0   . (A.14) Solving the last row b y the metho d of characteristic, w e obtain I = I   n, z , z ,  z − z   z − z   z − z   z − z    . (A.15) Substituting this bac k to the other t wo equations gives I = I   n,  z − z   z − z   z − z   z − z    , (A.16) and hence w e get the generating diﬀerence inv ariants n and I 1 =  z − z   z − z   z − z   z − z  . (A.17) F rom a ﬁrst glance, it lo oks diﬀeren t from the cross-ratio inv ariant (2.39), temporarily denoted by I 0 ; in fact, they are related via the relation I 0 + I 1 = I 0 I 1 . (A.18) One can similarly prov e that the I 1 [ z 1 , z 2 ] and I 2 [ z 1 , z 2 ] deﬁned by (2.47) (or (2.46)) are indeed in v arian t corresp onding to the PGL(3)-action b y showing that they satisfy the follo wing system of linear PDEs pr X i I  n, z 1 , z 2 , z 1 , z 2 , z 1 , z 2 , z 1 , z 2 , z 1 , z 2  = 0 . (A.19) Here, the eight inﬁnitesimal generators are in the same form as the contin uous PGL(3)- action analysed in Appendix A.1 but with discrete prolongations. Substitution of the functions (2.47) into the abov e system immediately shows that b oth of them are in v arian t, and co eﬃcien t matrix of the abov e system is of maximal rank 8. Hence, these tw o functions and n serve as generating inv ariants of the PGL(3)-action about ( n, z 1 , z 2 ). All other diﬀerence inv ariants are functions of generating inv ariants and their shifts. App endix B PGL( N ) -in v arian ts from linear sp ectral prob- lems In this app endix, we show ho w the metho d introduced in Section 2 for obtaining inv ari- an ts from linear sp ectral problems can be generalized to PGL( N ) actions, in b oth the diﬀeren tial and diﬀerence settings. 35 B.1 The diﬀeren tial case Consider an N th-order linear sp ectral problem ∂ N x φ + u N − 2 ∂ N − 2 x φ + · · · + u 1 ∂ x φ + u 0 φ = λφ , (B.1) and its independent solutions φ 1 , φ 2 , . . . , φ N . The natural GL( N )-action on solutions ( φ 1 , φ 2 , . . . , φ N ) 7→ ( φ ′ 1 , φ ′ 2 , . . . , φ ′ N ) = ( φ 1 , φ 2 , . . . , φ N ) M (B.2) induces the PGL( N )-action on the inhomogeneous co ordinates z i = φ i φ N , i = 1 , 2 , . . . , N − 1 (B.3) as follows z ′ i = N − 1 P j =1 m j i z j + m N i N − 1 P j =1 m j N z j + m N N , i = 1 , 2 , . . . , N − 1 , where det M  = 0 . (B.4) W e in tro duce the notations φ = φ (0) , ∂ x φ = φ (1) , ∂ 2 x φ = φ (2) and so forth. Note that they stand for shifts in the discrete case. W riting the equations for φ = φ N and φ i = z i φ ( i = 1 , 2 , . . . , N − 1), w e obtain the following system: φ ( N ) + u N − 2 φ ( N − 2) + · · · + u 1 φ (1) + ( u 0 − λ ) φ = 0 , (B.5a) C N − 1 N z (1) i φ ( N − 1) + N − 2 X k =0 C k N z ( N − k ) i + N − 2 X l = k +1 C k l u l z ( l − k ) i ! φ ( k ) = 0 . (B.5b) Here, we use the notation C l k = k ! l !( k − l )! . (B.6) Dividing b y φ on b oth sides of (B.5b), the resulting equations can b e written in matrix form             z ( N − 1) 1 z ( N − 2) 1 . . . z (1) 1 . . . . . . . . . . . . z ( N − 1) i z ( N − 2) i . . . z (1) i . . . . . . . . . . . . z ( N − 1) N − 1 z ( N − 2) N − 1 . . . z (1) N − 1             Φ = −             z ( N ) 1 . . . z ( N ) i . . . z ( N ) N − 1             , (B.7) where the v ector Φ is given b y ( l = N − 2 , N − 1 , . . . , 1) Φ = C 1 N φ (1) φ , . . . , C N − l N φ ( N − l ) φ + N − 2 − l X k =1 C k k + l u k + l φ ( k ) φ + u l , . . . ! ⊺ . (B.8) Deﬁning z = ( z 1 , z 2 , . . . , z N − 1 ) ⊺ , the system (B.7) can be solved b y Cramer’s rule. In particular, we obtain a higher-order Hopf–Cole-t yp e transformation φ (1) φ = − 1 N   z ( N ) , z ( N − 2) , . . . , z (1)     z ( N − 1) , z ( N − 2) , . . . , z (1)   . (B.9) 36 In fact, φ can b e solved from (B.9): φ = c    z ( N − 1) , z ( N − 2) , . . . , z (1)    − 1 N (B.10) and without loss of generality , the integration constan t can b e c hosen as c = 1. By doing so, we are released from deriving φ ( N ) /φ from the recursive relation for large N : φ ( N ) φ = ∂ x φ ( N − 1) φ ! + φ (1) φ φ ( N − 1) φ . (B.11) The function φ and its deriv atives are then substituted bac k to the linear system (B.5), whose solution ab out the p otentials will give us ( N − 1)-n umber of diﬀeren tial inv ariants. In fact, the system (B.5) is now equiv alen t to com bing solution of (B.7) and (B.5a), which can b e written in matrix form for the potentials as A               u N − 2 u N − 3 . . . u 1 u 0 − λ               = −                | z ( N − 1) , z ( N ) , z ( N − 3) ,..., z (1) | | z ( N − 1) , z ( N − 2) ,..., z (1) | + C 2 N φ (2) φ | z ( N − 1) , z ( N − 2) , z ( N ) ,..., z (1) | | z ( N − 1) , z ( N − 2) ,..., z (1) | + C 3 N φ (3) φ . . . | z ( N − 1) , z ( N − 2) ,..., z (2) , z ( N ) | | z ( N − 1) , z ( N − 2) ,..., z (1) | + C N − 1 N φ ( N − 1) φ φ ( N ) φ                , (B.12) where the coe ﬃcien t matrix is a triangular matrix A =                     1 0 0 . . . 0 0 0 C 1 N − 2 φ (1) φ 1 0 . . . 0 0 0 C 2 N − 2 φ (2) φ C 1 N − 3 φ (1) φ 1 . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C N − 3 N − 2 φ ( N − 3) φ C N − 4 N − 3 φ ( N − 4) φ C N − 5 N − 4 φ ( N − 5) φ . . . C 1 2 φ (1) φ 1 0 φ ( N − 2) φ φ ( N − 3) φ φ ( N − 4) φ . . . φ (2) φ φ (1) φ 1                     . (B.13) The ab o ve system can b e solved via Cramer’s rule or via the following recursiv e relation: u N − 2 = −   z ( N − 1) , z ( N ) , z ( N − 3) , . . . , z (1)     z ( N − 1) , z ( N − 2) , . . . , z (1)   − C 2 N φ (2) φ , u k = −   z ( N − 1) , . . . , z ( k +1) , z ( N ) , z ( k − 1) , . . . , z (1)     z ( N − 1) , z ( N − 2) , . . . , z (1)   − C N − k N φ ( N − k ) φ − N − 2 − k X l =1 C l k + l u k + l φ ( l ) φ , k = N − 3 , N − 4 , . . . , 1 , u 0 − λ = − φ ( N ) φ − N − 2 X l =1 u l φ ( l ) φ . (B.14) 37 Due to the indep endence of the p otentials u N − 2 , u N − 1 , . . . , u 0 , these inv ariants are func- tionally indep enden t from each other. Remark B.1 It is cle ar that the highest or ders of derivatives involve d in the invariants (B.14) ar e not the same, but r ather it incr e ases one e ach time. F or instanc e, u N − 2 includes derivatives of z up to the or der N + 1 , while u N − 3 includes derivatives of z up to the or der N + 2 . However, these invariants c an b e r e-arr ange d to a set of fundamental invariants, which al l dep end on derivatives of z up to the or der N + 1 ; this c an b e done by taking the r e cursive r elation (B.11) into c onsider ation. F or instanc e, a new invariant C 2 N u N − 3 − C 3 N ∂ x u N − 2 , (B.15) that involves derivatives up to or der N + 1 , c an b e use d to r eplac e u N − 3 , that involves derivatives up to or der N + 2 . The case N = 2 is straigh tforward. Let us consider N = 3 as an example to illustrate ho w the inv ariants (2.14) can b e obtained accordingly . F or the third-order sp ectral problem (2.7), the system (B.7) b ecomes   z (2) 1 z (1) 1 z (2) 2 z (1) 2     3 φ (1) φ 3 φ (2) φ + u   = −   z (3) 1 z (3) 2   . (B.16) Solving it, w e obtain the Hopf–Cole transformation (2.12) and u = −   z (2) , z (3)     z (2) , z (1)   − 3 φ (2) φ . (B.17) Solving for φ from the Hopf–Cole transformation leads to φ =    z (2) , z (1)    − 1 3 , (B.18) whic h is substituted bac k to u . This gives the ﬁrst inv ariant S 1 [ z 1 , z 2 ] in (2.14). The other one, i.e. − 3( v − λ ) + u x = S 2 [ z 1 , z 2 ], is obtained by substituting the results bac k to the sp ectral problem (2.7). B.2 The diﬀerence case Consider an N th-order linear diﬀerence equation of the form Λ φ = λφ , Λ = T N + h N − 1 T N − 1 + · · · + h 1 T + h 0 , (B.19) with λ , h 0 , h 1 , . . . , h N − 1 a collection of p oten tial functions whic h can b e realised as diag- onal matrices. W e hav e a natural GL( N )-action on ( φ 1 , φ 2 , . . . , φ N ), i.e. the v ector of indep enden t solutions. Introducing z i = φ i φ N , i = 1 , 2 , . . . , N − 1 , (B.20) and denoting φ = φ N , we get the set of equations: φ ( N ) + h N − 1 φ ( N − 1) + · · · + h 1 φ (1) − ( λ − h 0 ) φ = 0 , (B.21a) z ( N ) i φ ( N ) + h N − 1 z ( N − 1) i φ ( N − 1) + · · · + h 1 z (1) i φ (1) − ( λ − h 0 ) z i φ = 0 , (B.21b) 38 where the superscript denotes the order of the shift: z ( j ) i = T j z i . The ab o ve system can b e rewritten in matrix form         1 1 . . . 1 z ( N ) 1 z ( N − 1) 1 . . . z (1) 1 . . . . . . . . . . . . z ( N ) N − 1 z ( N − 1) N − 1 . . . z (1) N − 1                  φ ( N ) φ h N − 1 φ ( N − 1) φ . . . h 1 φ (1) φ          = ( λ − h 0 )        1 z 1 . . . z N − 1        . (B.22) Let us in tro duce the notation z = (1 , z 1 , z 2 , . . . , z N − 1 ) ⊺ temp orarily . The system can b e solv ed using Cramer’s rule, namely φ ( N ) φ = ( λ − h 0 )   z , z ( N − 1) , . . . , z (1)     z ( N ) , z ( N − 1) , . . . , z (1)   , φ ( i ) φ = λ − h 0 h i   z ( N ) , , . . . , z ( i +1) , z , z ( i − 1) , . . . , z (1)     z ( N ) , z ( N − 1) , . . . , z (1)   , i = 1 , 2 , . . . , N − 1 . (B.23) By eliminating φ and its shifts using φ ( i ) φ · φ φ ( i − 1) ·  φ φ (1)  ( i − 1) = 1 , i = 2 , 3 , . . . , N , (B.24) w e obtain the following ( N − 1)-num b er of diﬀerence in v arian ts: h i − 1 h ( i − 1) 1 ( λ − h 0 ) h i =   z ( N ) , . . . , z ( i ) , z , z ( i − 2) , . . . , z (1)     z ( N ) , . . . , z ( i +1) , z , z ( i − 1) , . . . , z (1)     z ( N ) , z ( N − 1) , . . . , z (2) , z     z ( N ) , z ( N − 1) , . . . , z (2) , z (1)   ! ( i − 1) (B.25) for i = 2 , 3 , . . . , N , where w e temp orarily deﬁne h N = 1. Similarly to the contin uous case, e ac h inv ariant of (B.25) is written in terms of diﬀerent n umbers of v ariables, and the highest order of c hanges increases b y 1 as the index i increases by 1. A set of diﬀerence in v arian ts, that dep end on the same num b er of v ariables, i.e. z , z (1) , . . . , z ( N ) , z ( N +1) , can b e deriv ed as follows: h 1 h (1) 1 ( λ − h 0 ) h 2 =   z ( N ) , . . . , z (3) , z (2) , z     z ( N ) , . . . , z (3) , z , z (1)     z ( N ) , z ( N − 1) , . . . , z (2) , z     z ( N ) , z ( N − 1) , . . . , z (2) , z (1)   ! (1) , h i − 1 h (1) i − 1 h i h (1) i − 2 = h i − 1 h ( i − 1) 1 ( λ − h 0 ) h i . h i − 2 h ( i − 2) 1 ( λ − h 0 ) h i − 1 ! (1) =   z ( N ) , . . . , z ( i ) , z , z ( i − 2) , . . . , z (1)     z ( N ) , . . . , z ( i +1) , z , z ( i − 1) , . . . , z (1)     z ( N ) , . . . , z ( i ) , z , z ( i − 2) , . . . , z (1)     z ( N ) , . . . , z ( i − 1) , z , z ( i − 3) , . . . , z (1)   ! (1) for i = 3 , 4 , . . . , N . App endix C Darb oux–Crum form ulae The Darb oux transformations discussed in Sections 3 generate exact discretisations and rev eal the self-dual structure underlying the BSQ equations. Here we presen t the asso ciated Darb oux–Crum form ulae, along the line of [71] where the analogous form ulae for the KdV case of systems w ere developed, whic h serv e as explicit formulae for m ulti-soliton solutions of the BSQ equations. 39 Theorem C.1 (Contin uous Darb oux–Crum formulae) Consider the c ontinuous sp e c- tr al pr oblem (2.7) . Assume ther e exist N line arly indep endent solutions φ j , j = 1 , 2 , . . . , N , of the sp e ctr al pr oblem L φ = λφ at λ = α j r esp e ctively, then an N -step Darb oux tr ansfor- mation amounts to the map { φ, u, v } 7→ { φ [ N ] , u [ N ] , v [ N ] } : φ [ N ] = W ( φ 1 , . . . , φ N , φ ) W ( φ 1 , . . . , φ N ) , (C.1) wher e W ( ∗ , . . . , ∗ ) denotes the Wr onskian, and u [ N ] = u − 3 s ′ 1 , v [ N ] = v + N u ′ + 3( s 1 s ′ 1 − s ′′ 1 − s ′ 2 ) , (C.2) wher e s 1 , s 2 ar e deﬁne d as s 1 = − | Φ , . . . , Φ ( N − 2) , Φ ( N ) | W ( φ 1 , . . . , φ N ) , s 2 = | Φ , . . . , Φ ( N − 3) , Φ ( N − 1) , Φ ( N ) | W ( φ 1 , . . . , φ N ) , (C.3) with Φ = ( φ 1 , . . . , φ N ) ⊺ , and the sup erscript ( n ) me aning the n th-or der derivative. T o b e c onsistent with Se ction 3, note that the primes in (C.2) denote derivatives. Pro of: the action of an N -step Darb oux transformation on φ using φ j , j = 1 , 2 , . . . , N , yields φ [ N ] = φ ( N ) + s 1 φ ( N − 1) + s 2 φ ( N − 2) + · · · + s N φ , (C.4) where the coe ﬃcien t functions s j are determined b y the linear system    φ 1 φ (1) 1 . . . φ ( N − 1) 1 . . . . . . . . . . . . φ N ϕ (1) N . . . φ ( N − 1) N       s N . . . s 1    = −    φ ( N ) 1 . . . φ ( N ) N    , (C.5) whic h is the consequence of φ [ N ] | φ = φ j = 0. The expressions of u [ N ] and v [ N ] are obtained b y equating the co eﬃcients of L [ N ] φ [ N ] = λφ [ N ] with L [ N ] := ∂ 3 x + u [ N ] ∂ x + v [ N ] . Theorem C.2 (Discrete Darb oux–Crum formulae) Consider the discr ete sp e ctr al pr ob- lem (2.42) . Assume ther e exist M line arly indep endent solutions φ j , j = 1 , 2 , . . . , M , of the sp e ctr al pr oblem Λ φ = λφ at λ = β j r esp e ctively, then a M -step Darb oux tr ansformation amounts to the map { φ, h , g } 7→ { φ ⟨ M ⟩ , h ⟨ M ⟩ , g ⟨ M ⟩ } : φ ⟨ M ⟩ = C ( φ 1 , . . . , φ M , φ ) C ( φ 1 , . . . , φ M ) , (C.6) wher e C ( ∗ , . . . , ∗ ) denotes the Casor ati determinant, and h ⟨ M ⟩ = h [ M ] − σ 1 , g ⟨ M ⟩ = g [ M ] − σ 2 + s 1 σ 1 + s 1 h [ M − 1] − s 1 h [ M ] , (C.7) wher e σ ℓ = s ℓ − s ℓ , ℓ = 1 , 2 , and s ℓ ar e deﬁne d as s 1 = − | Φ , . . . , Φ [ M − 2] , Φ [ M ] | C ( φ 1 , . . . , φ M ) , s 2 = | Φ , . . . , Φ [ M − 3] , Φ [ M − 1] , Φ [ M ] | C ( φ 1 , . . . , φ M ) , (C.8) with Φ = ( φ 1 , . . . , φ M ) ⊺ , and the sup erscript [ n ] me aning the n th-or der shift. Pro of: The pro of is in complete analogy with the contin uous case. 40 App endix D Elimination of P , Q in (4.44) and connected v ec- torial form W e rewrite the system (4.44), which is linear in P , Q , as follows P s = N P + K Q , (D.1a) Q t = M Q + L P , (D.1b) P t + Q s = A P + B Q , (D.1c) with the abbreviations: N := 1 − n s + | z t , z ss | | z t , z s | , M := 1 − m t + | z s , z tt | | z s , z t | , (D.2a) K := s t | z s , z ss | | z s , z t | , L := t s | z t , z tt | | z t , z s | , (D.2b) A := 2 | z t , z st | | z t , z s | , B := 2 | z s , z st | | z s , z t | . (D.2c) W e no w eliminate the quan tities P and Q as follows. First, from (D.1b) and (D.1c), using ( Q t ) s = ( Q s ) t w e ﬁnd (after back-substituting the expressions for Q t , Q s and P s from the system): ( M Q + LP ) s = ( AP + B Q − P t ) t ⇒ P tt = ( A + M ) P t + [ A t − L s + ( B − N ) L − AM ] P + ( B t − M s − K L ) Q . (D.3) Second, from (D.1a) b y diﬀeren tiating with resp ect to t and bac k-substituting Q t from (D.1b), we get P st = N P t + ( N t + K L ) P + ( K t + M K ) Q . (D.4) Using now the equalit y ( P tt ) s = ( P st ) t through the ab o ve expressions for P tt and P st , and p erforming the v arious bac k-substitutions for Q s , Q t and P s from (D.1) and the already obtained expressions for P tt and P st , we arriv e at: (2 N t + B t − 2 M s − A s ) P t + h N tt + ( K L ) t + LK t − ( A + M ) N t + L ss − A st + ( AM ) s − (( B − N ) L ) s + A ( M s − B t ) i P + h K tt + ( M K ) t − AK t − K ( A t − L s ) + M ss + ( K L ) s − B st + ( B − N )( M s − B t ) i Q = 0 . (D.5) Here the coe ﬃcien t of P t v anishes iden tically , using the expressions (D.2): 2 N t + B t = 2 ∂ t ( ∂ s log | z t , z s | ) = 2 M s + A s . (D.6) Setting the co eﬃcients of P and Q in the remaining terms equal to zero, we get the tw o equations: N tt + L ss − A st + ( K L ) t + ( AM ) s + AM s + LK t − ( A + M ) N t − (( B − N ) L ) s − AB t = 0 , (D.7a) M ss + K tt − B st + ( K L ) s + ( B N ) t + B N t + K L s − ( B + N ) M s − (( A − M ) K ) t − A s B = 0 , (D.7b) 41 where (D.7) is used to bring the co eﬃcien ts in the symmetric form. First, by substituting the expressions (D.2) in to (D.7a) and (D.7b), we obtain the equations (4.45) expressed in terms of the 2 × 2 determinants of the deriv ativ es of the v ector z . Second, to obtain the v ectorial equations (4.67) from the system (D.7), we note that the terms with the highest deriv ativ e z sstt in (D.7a) and (D.7b) are  t s − 1  | z t , z ttss | | z t , z s | and  s t − 1  | z s , z sstt | | z s , z t | , (D.8) resp ectiv ely . 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F rank Nijhoﬀ Sc ho ol of Mathematics, Universit y of Leeds, Leeds LS2 9JT, United Kingdom f.w.nijhoff@leeds.ac.uk Lin yu P eng Departmen t of Mechanical Engineering, Keio Universit y , Y okohama 223-8522, Japan l.peng@mech.keio.ac.jp Cheng Zhang Departmen t of Mathematics, Shanghai Univ ersity , Shanghai 200444, China Newtouc h Cen ter for Mathematics of Shanghai Universit y , Shanghai 200444, China ch.zhang.maths@gmail.com Da-jun Zhang Departmen t of Mathematics, Shanghai Univ ersity , Shanghai 200444, China Newtouc h Cen ter for Mathematics of Shanghai Universit y , Shanghai 200444, China djzhang@staff.shu.edu.cn 46

$\mathrm{PGL}(3)$-invariant integrable systems from factorisation of linear differential and difference operators

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