Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions

Classificat ion of in t egrable t w o-comp onen t Hamiltonian systems of h ydro d ynamic t yp e in 2 + 1 dimensions E.V. F erap on tov 1 , A.V. Odesskii 2 and N.M. Stoilo v 1 1 Departmen t of Mathematical Sciences Lough b orough Univ ersit y Lough b orough, Leicestershire LE11 3TU United Kingdom 2 Departmen t of Mathematics Bro c k Univ ersit y St. Catharines, On tario L2S 3A1 Canada e-mails: E.V.Ferapontov@lboro.ac.uk aodesski@brocku.ca N.M.Stoilov@lboro.ac.uk Abstract Hamiltonian systems of h ydro dynamic t ype o ccur in a wide r ange of a pplications including fluid dyna mics , the Whitham averaging proc e dur e and the theo r y o f F ro benius manifolds. In 1 + 1 dimensions, the req uiremen t of the in tegra bilit y of such systems b y the g eneralised ho dograph transfor m implies that int egrable Hamiltonia ns dep end on a certain num b er o f arbitrary functions of t wo v ariables. On the con trary , in 2 + 1 dimensions the requiremen t of the integrabilit y by the metho d of h ydr o dynamic reductions, whic h is a natural analogue of the genera lised ho dogr aph trans - form in higher dimensions, leads to finite-dimensiona l mo duli spaces of integrable Hamilto- nians. In this paper we classify in tegrable t w o-comp onent Ha miltonia n sy s tems of h ydro dy- namic type for all existing cla s ses o f differen tial-geometric Poisson br ac kets in 2 D, establish- ing a parametr is ation o f integrable Hamiltonians via elliptic/hypergeometr ic functions. Our approach is based o n the Go dunov-t yp e r epresent ation of Hamiltonian systems, and utilises a nov el construction of Go dunov’s systems in terms of genera lised hypergeo metric functions. MSC: 35L40, 35L65, 37K10. Keywords: Multi-dimensional Systems of Hydro dynamic Type, Hamiltonian Structure s , Disper sionless L a x P airs , Hydro dynamic Reductions, Godunov F orm, Generalised Hyp erge- ometric F unctions. 1 Con ten ts 1 In tro duction 3 2 Hamiltonian systems of t yp e (6) 7 2.1 Classification of integr able p oten tials . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Hamiltonian systems of t yp e (7) 10 3.1 Classification of integr able p oten tials . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Disp ersionless Lax pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Hamiltonian systems of t yp e (8) 16 4.1 Classification of integr able p oten tials . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Disp ersionless Lax pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Go dunov systems and generalized hypergeometric functions 25 5.1 Go duno v systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 Go duno v form of in tegrable quasilinear systems and generalized hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 5.3 Application to in tegrable p oten tials H(V, W) . . . . . . . . . . . . . . . . . . . . 28 6 App endix. The metho d of hydro dynamic reductions. Deriv ation of the inte- grabilit y conditions 30 2 1 In tro duction A 1 + 1 dimensional quasilinear system, u t = A ( u ) u x , is said to b e Hamiltonian if it can b e represen ted in the f orm u t = P h u where h ( u ) is the Hamiltonian densit y , h u is its gradien t, u = ( u 1 , . . . , u n ) , and P = P ij is a Hamiltonian op erator of differen tial-geometric t yp e, P ij = g ij ( u ) d dx + b ij k ( u ) u k x . It w as demonstrated in [10] that the co efficien ts g ij define a flat con trav arian t metric, and b i j k can b e expressed in terms of its Levi-C ivita connection via b ij k = − g is Γ j sk . This implies the existence of flat co ordinate s where the op erator P assumes a constan t-co efficien t form, P ij = ǫ i δ ij d dx . In the same co ordinates, the corresp onding matr ix A is the Hessian matrix of h. It w as conjectured b y Novik o v that a com bination of the Hamiltonian prop ert y with the diagonalisabi lit y of the matrix A implies the in tegrabilit y . This conjecture w as pro ved b y T s arev in [37] who established the linearisabilit y of such systems by the general ised hodograph transform. Hamilto nian systems arise in a wide range of application s in fluid dynamics and the Whitham a verag ing theory . The underlying differen tial-geometric structure mak e them the main building blo c k s of the theory of F robenius manifolds [12]. Let us note that the requiremen t of the diagonalisabilit y of the matrix A imp oses a complicated system of third order PDEs for the Hamiltonian densit y h whose general solution is k no wn to b e parametrised b y n ( n − 1) / 2 arbitrary functions of tw o v ariables. Th us, the mo duli space of inte grable Hamiltonians in 1 + 1 dimensions is essen tially infinite-dime nsional. W e refer to [10, 38, 11] for further discussion. In this pap er we provide a complete classification of in tegrable tw o-component 2 + 1 dimen- sional Hamilton ian sys tems of h ydro dynamic t yp e. Our approac h is based on the metho d of h ydro dynamic reductions whic h, in s ome sense, can b e view ed as a natural extension of the generalised ho dograph transform to higher dimensions. Our results demonstrate that the moduli spaces of in tegrable Hamiltonians in 2 + 1 dimensions are finite-dimen sional: the in tegrabil- it y conditions for the corresp onding Hamiltonian densities constitute o ver-de termined inv olutiv e system of finite type. Mo dulo natural equiv alence groups, this leads to finite lists of in tegrable Hamiltonians parametrised b y elliptic/h yp ergeometric functions. Th us, we consider 2 + 1 dimensional quasilinear systems u t = A ( u ) u x + B ( u ) u y (1) whic h can b e represen ted in the H amilton ian form, u t = P h u . (2) Here h ( u ) is the Hamiltonian densit y , h u is its gradien t, and P = P ij is a Hamiltoni an op erator of differen tial-geometric t yp e, P ij = g ij ( u ) d dx + b ij k ( u ) u k x + ˜ g ij ( u ) d dy + ˜ b ij k ( u ) u k y . Op erators of this form are generated b y a pair of metrics g ij , ˜ g ij , see [10, 25, 26 ] for the general theory and classification results. The main difference from t he one-dimensional situation is that, 3 although both metrics g ij and ˜ g ij m ust necessarily b e flat, they can no longer b e reduced to constan t coefficien t f orm s im ultaneously: there exist obstruction tensors. The tw o-componen t situation is understo o d completely . One has only three t yp es of Hamiltonian op erators: the first t wo of them can be reduced to constan t-co efficien t forms, P =  d/dx 0 0 d/dy  (3) and P =  0 d/dx d/dx d/dy  , (4) while the third one is essentiall y non-constan t, P =  2 v w w 0  d dx +  0 v v 2 w  d dy +  v x v y w x w y  . (5) Here v , w are comp onen ts of the vecto r u , and h u = ( h v , h w ) t . Hamiltonian systems generated b y the op erators (3) - (5) tak e the form v t = ( h v ) x , w t = ( h w ) y , (6) v t = ( h w ) x , w t = ( h v ) x + ( h w ) y , (7) and v t = (2 v h v + w h w − h ) x + ( v h w ) y , w t = ( w h v ) x + (2 w h w + v h v − h ) y , (8) resp ectiv ely . Our main result is a description of Hamiltonian densities h ( v , w ) f or whic h the corresp onding s ystems (6) - (8) are in tegrable b y the method of h ydro dynamic reductions as prop osed in [13]. Let us p oin t out that the integr abilit y conditions for the general class of t wo - comp onent systems (1) were deriv ed in [14] in the co ordinates where the first matrix A is diagonal. It wa s demonstrated in [27, 29] that, again in sp ecial co ordinates, the matrices A and B can be parametrised by h y pergeometric functions. Ho w ever, it is not clear ho w to isolate Hamiltonian cases within these general descriptions. In this pap er w e adopt a straigh tforw ard approac h and deriv e the in tegrabilit y conditions b y directly applying the metho d of h ydro dynamic reductions to the systems (6) - (8). F or Hamilto nian sy stems of type (6) this was done in [13]. Th us, in the App endix w e illustrate the metho d of h ydro dynamic reductions by calculati ng the in tegrabilit y conditions for equations of type (7). Applied to equations (6) - (8), this metho d results in in volutiv e systems of fourth order PDEs for the Hamiltonian densities h ( v , w ) whic h are quite complicated in general. It was observed that these sy stems s implify considerably under the Legendre transformatio n h ( v , w ) → H ( V , W ) defined as V = h v , W = h w , H = v h v + w h w − h, H V = v , H W = w . In the new v ariables, equations (6) - (8) take the so-called Go duno v form, ( H V ) t = V x , ( H W ) t = W y , (9) ( H V ) t = W x , ( H W ) t = V x + W y , (10) and ( H V ) t = ( V H V + H ) x + ( W H V ) y , ( H W ) t = ( V H W ) x + ( W H W + H ) y , (11) 4 resp ectiv ely . All our classification results will b e form ulated in terms of the Legendre-transformed Hamiltonian densities H ( V , W ) . W e recall that a system (1) is said to b e in Go duno v f orm [21] if, for appropriate p oten tials F α ( u ) , α = 0 , 1 , 2 , it p ossess es a conserv ative represen tation ( F 0 ,i ) t + ( F 1 ,i ) x + ( F 2 ,i ) y = 0 , F α,i = ∂ F α /∂ u i . In particular, in the case (11) one has F 0 = − H , F 1 = V H , F 2 = W H . This represen tation will b e utilised in Sect. 5 to provide a parametrisation of the generic in tegrable p oten tial of t yp e (11) b y generalised h yp ergeometric f unctions. Let us first review the kno wn results for systems of type (9). Here the int egrabilit y conditions tak e the form H V W H V V V V = 2 H V V V H V V W , H V W H V V V W = 2 H V V V H V W W , H V W H V V W W = H V V W H V W W + H V V V H W W W , H V W H V W W W = 2 H V V W H W W W , H V W H W W W W = 2 H V W W H W W W , see [13]. These equations are in in volutio n and can b e solv ed in closed form. Up to the action of a natural equiv alence group, this pro vides a complete list of in tegrable p oten tials. Theorem 1 [16] T he generic inte gr able p otential of typ e (9) is given by the formula H = Z ( V + W ) + ǫZ ( V + ǫW ) + ǫ 2 Z ( V + ǫ 2 W ) wher e ǫ = e 2 π i/ 3 and Z ′′ ( s ) = ζ ( s ) . Her e ζ is the W eierstr ass zeta-function, ζ ′ = − ℘, ( ℘ ′ ) 2 = 4 ℘ 3 − g 3 . De gener ations of this solution c orr esp ond to H = ( V + W ) log ( V + W ) + ǫ ( V + ǫW ) log ( V + ǫW ) + ǫ 2 ( V + ǫ 2 W ) log( V + ǫ 2 W ) , H = 1 2 V 2 ζ ( W ) , H = V 2 2 W , H = ( V + W ) ln( V + W ) , as wel l as the fol lowing p olynomial p otentials: H = V 2 W 2 , H = V W 2 + α 5 W 5 , H = V W + 1 6 W 3 . W e refer to Sect. 2 for further details. In Sect. 3 we demonstrate that f or systems of type (10) the integ rabilit y conditions tak e the form H V V V V = 2 H 2 V V V H V V , H V V V W = 2 H V V W H V V V H V V , H V V W W = 2 H 2 V V W H V V , (12) H V W W W = 3 H V W W H V V W − H W W W H V V V H V V , H W W W W = 6 H 2 V W W − 4 H W W W H V V W H V V . 5 These equations are also in in v olution, and can b e solv ed in closed form. Up to the action of a natural equiv alence group, w e obtain a complete list of in te grable p otent ials. Theorem 2 The generic inte gr able p otential of typ e (10) is given by the formula H = V ln V σ ( W ) wher e σ is the W eierstr ass sigma-function: σ ′ /σ = ζ , ζ ′ = − ℘, ℘ ′ 2 = 4 ℘ 3 − g 3 . Its de gener ations c orr esp ond to H = V ln V W , H = V ln V , H = V 2 2 W + αW 7 , as wel l as the fol lowing p olynomial p otentials: H = V 2 2 + V W 2 2 + W 4 4 , H = V 2 2 + W 3 6 . F urther details, as wel l as the disp ersionless Lax pairs corresp onding to in tegrable p oten tials from Theorem 2, are pro vided in Sect. 3. Remark. F or H = V ln V the eq uations (10 ) take the form V t = V W x , V x = − W y , implying the Bo yer-Finle y equations for V : V xx + (ln V ) ty = 0 . Simila rly , the c hoice H = V 2 2 + W 3 6 results in the equations V t = W x , V x = W W t − W y , implying the dKP equation for W : W xx = ( W W t − W y ) t . These Hamiltonian represen tations ha ve appeared previously in the literature, see e.g. [4]. The case (11) turns out to b e considerably more complicated. The in tegrabilit y conditions constitute an in volutiv e system of fourth order PDEs f or the p oten tial H which is not presen ted here due to its complexit y (see Sect. 4). Ho wev er, it p ossesses a remark able S L (3 , R ) -in v ariance whic h reflects the in v ariance of the Hamiltonian formalism (8 ) under linear transformations of the indep enden t v ariables x, y , t . Based on this in v ariance, w e were able to classify in tegrable p oten tials. Theorem 3 The generic inte gr able p otential H ( V , W ) of typ e (11) is given by the series H = 1 W g ( V )  1 + 1 g 1 ( V ) W 6 + 1 g 2 ( V ) W 12 + ...  . Her e g ( V ) satisfies the fourth or der ODE, g ′′′′ (2 gg ′ 2 − g 2 g ′′ ) + 2 g 2 g ′′′ 2 − 20 g g ′ g ′′ g ′′′ + 16 g ′ 3 g ′′′ + 18 g g ′′ 3 − 18 g ′ 2 g ′′ 2 = 0 , whose gener al s olution c an b e r epr esente d in p ar ametric for m as g = w 2 ( t ) , V = w 1 ( t ) w 2 ( t ) , wher e w 1 ( t ) and w 2 ( t ) ar e two line arly indep endent solutions of the hyp er ge ometric e quation t (1 − t ) d 2 w /dt 2 − 2 9 w = 0 . The c o efficients g i ( V ) ar e c ertain explicit expr essions in terms of g ( V ) , e.g., g 1 ( V ) = gg ′′ − 2 g ′ 2 and so on. De gener ations of this solution c orr esp ond to H = 1 W g ( V ) , H = 1 W V , H = V − W log W, H = V − W 2 / 2 . 6 W e refer to Sect. 4 for further details. In Sect. 5 w e pro vide a parametrisation of the generic in tegrable p oten tial H ( V , W ) b y generalised hypergeometric functions: Theorem 4 The generic inte gr able p otential H ( V , W ) of typ e (11) c an b e p ar ametrise d by gen- er alise d hyp er ge ometric functions, H = G  u 1 − 1 u 2 − 1  , V = h 1 h 0 , W = h 2 h 0 . (13) Her e G ′ ( t ) = 1 t 2 / 3 ( t − 1) 2 / 3 and h 0 , h 1 , h 2 ar e thr e e line arly indep endent solution of the hyp er ge o- metric system u 1 (1 − u 1 ) h u 1 ,u 1 − 4 3 u 1 h u 1 − 2 9 h = 2 3 u 1 ( u 1 − 1) u 1 − u 2 h u 1 + 2 3 u 2 ( u 2 − 1) u 2 − u 1 h u 2 , u 2 (1 − u 2 ) h u 2 ,u 2 − 4 3 u 2 h u 2 − 2 9 h = 2 3 u 1 ( u 1 − 1) u 1 − u 2 h u 1 + 2 3 u 2 ( u 2 − 1) u 2 − u 1 h u 2 , h u 1 ,u 2 = 2 3 h u 2 − h u 1 u 2 − u 1 , which c an b e viewe d as a natur al two-dimensional gener alisati on of the hyp er ge ometric e quation t (1 − t ) d 2 w /dt 2 − 2 9 w = 0 . This parametrisation is based on a nov el construction of in tegrable quasilinear systems in Go- duno v’s form in terms of generalised h yp ergeometri c functions whic h builds on [27, 29 ]. W e b eliev e that this construction is of indep enden t inter est, and can b e applied to a whole v ariety of similar classification problems. The details are provided in Sect. 5 (Theorem 5). Our results lead to the followin g general observ ations: (i) The mo duli spaces of in tegrable Hamiltonian s in 2 D are finite dimensional . (ii) In the tw o-componen t case, the actions of the natural eq uiv alence groups on the mo duli spaces of in tegrable Hamilto nians p ossess op en orbits. This leads to a remark able conclusion that f or any tw o-componen t Poi sson brack et in 2D there exis ts a unique ‘generic’ integr able Hamiltonian, all other cases can b e obtained as its appropriate degenerations. W e anticip ate that the results of this pap er will find applications in the theory of infinite- dimensional F rob enius manifolds, see [8] for the first steps in this direction. 2 Hamiltoni an systems of t yp e (6) In this section w e review the classification of integ rable systems of the form (6),  v w  t =  d/dx 0 0 d/dy   h v h w  , or, explicitly , v t = ( h v ) x , w t = ( h w ) y . 7 The in tegrabilit y conditions w ere first deriv ed in [13] based on the metho d of hydrodynamic reductions. These conditions constitute a system of fourth order PDEs for the Hamiltonian densit y h ( v , w ) , h vw ( h 2 vw − h vv h w w ) h vv vv = 4 h vw h vv v ( h vw h vv w − h vv h vw w ) +3 h vv h vw h 2 vv w − 2 h vv h w w h vv v h vv w − h vw h w w h 2 vv v , h vw ( h 2 vw − h vv h w w ) h vv vw = − h vw h vv v ( h vv h w ww + h w w h vv w ) +3 h 2 vw h 2 vv w − 2 h vv h w w h vv v h vw w + h 2 vw h vv v h vw w , h vw ( h 2 vw − h vv h w w ) h vv w w = 4 h 2 vw h vv w h vw w − h vv h vv w ( h vw h w ww + h w w h vw w ) − h w w h vv v ( h vw h vw w + h vv h w ww ) , h vw ( h 2 vw − h vv h w w ) h vw w w = − h vw h w ww ( h w w h vv v + h vv h vw w ) +3 h 2 vw h 2 vw w − 2 h vv h w w h w ww h vv w + h 2 vw h w ww h vv w , h vw ( h 2 vw − h vv h w w ) h w ww w = 4 h vw h w ww ( h vw h vw w − h w w h vv w ) +3 h w w h vw h 2 vw w − 2 h vv h w w h w ww h vw w − h vw h vv h 2 w ww . (14) This system is in in v olution, and its solution space is 10 -dimensional. Eqs. (14) can b e repre- sen ted in compact form as sd 4 h = d 3 hds + 3 h vw (2 dh v dh w − h vw d 2 h ) d et( dn ) where s = h 2 vw ( h vv h w w − h 2 vw ) , d k h denotes k -th symmetric differen tial of h , and n is the Hessian matrix of h . The con tact symmetry group of the system (14) is also 10 -dimensional, consisting of the 8 -parameter group of Lie-p oin t symmetries, v → av + b, w → cw + d, h → αh + β v + γ w + δ, along with the t wo purely cont act infinitesimal generators, − Ω h v ∂ ∂ v − Ω h w ∂ ∂ w + (Ω − h v Ω h v − h w Ω h w ) ∂ ∂ h + (Ω v + h v Ω h ) ∂ ∂ h v + (Ω w + h w Ω h ) ∂ ∂ h w , with the generating functions Ω = h 2 v and Ω = h 2 w , resp ectivel y . I t was observ ed in [13] that the in tegrabilit y conditions simplify under the Legendre transformation, V = h v , W = h w , H = v h v + w h w − h, H V = v , H W = w , whic h brings Eqs. (6) in to the Go duno v form (9), ( H V ) t = V x , ( H W ) t = W y . 8 The int egrabilit y conditions (14) simplify to H V W H V V V V = 2 H V V V H V V W , H V W H V V V W = 2 H V V V H V W W , H V W H V V W W = H V V W H V W W + H V V V H W W W , H V W H V W W W = 2 H V V W H W W W , H V W H W W W W = 2 H V W W H W W W . (15) This s y stem is also in inv olution, and can b e represen ted in compact form as S d 4 H = d 3 H dS + 6 H V W dV dW det( dN ) where S = H 2 V W , and N is the Hess ian matrix of H . The system (15) is in v arian t under a 10 -param eter group of Lie-point symmetries, V → aV + b, W → cW + d, H → αH + β V 2 + γ W 2 + µV + ν W + δ. Th us, b oth cont act symmetry generators of the s ystem (14) are mapp ed b y the Legendre trans- formation to p oin t symmetries of the sys tem (15). One can show that the action of the s y mmetry group on the mo duli space of solutions of the system (15) p ossesses an op en orbit. The classifi- cation of in tegrable p oten tials H ( V , W ) is p erformed modulo this equiv alence. 2.1 Classifica tion of in t egrable p otentia ls The pap er [16] pro vides a complete list of in tegrable p oten tials: Theorem 1 T he generic inte gr able p otential of typ e (9) is given by the formula H = Z ( V + W ) + ǫZ ( V + ǫW ) + ǫ 2 Z ( V + ǫ 2 W ) wher e ǫ = e 2 π i/ 3 and Z ′′ ( s ) = ζ ( s ) . Her e ζ is the W eierstr ass zeta-function, ζ ′ = − ℘, ( ℘ ′ ) 2 = 4 ℘ 3 − g 3 . De gener ations of this solution c orr esp ond to H = ( V + W ) log ( V + W ) + ǫ ( V + ǫW ) log ( V + ǫW ) + ǫ 2 ( V + ǫ 2 W ) log( V + ǫ 2 W ) , H = 1 2 V 2 ζ ( W ) , H = V 2 2 W , H = ( V + W ) ln( V + W ) , as wel l as the fol lowing p olynomial p otentials: H = V 2 W 2 , H = V W 2 + α 5 W 5 , H = V W + 1 6 W 3 . The corresponding systems (9) p ossess disp ersionless Lax pairs, see [16] for further details. 9 3 Hamiltoni an systems of t yp e (7) In this section w e consider Hamiltonian systems of the form (7),  v w  t =  0 d/dx d/dx d/dy   h v h w  , or, explicitly , v t = ( h w ) x , w t = ( h v ) x + ( h w ) y . Let us first men tion tw o w ell-kno wn examples whic h fall in to this class. Example 1. The Hamiltonian h = w 2 2 + e v generates the sys tem v t = w x , w t = e v v x + w y . Under the substitution v = u x , w = u t these equations reduce to u tt − u ty = e u x u xx , whic h is an equiv alen t form of the Boy er-Finley equation [6]. Example 2. The Hamiltonian h = v 2 2 + 2 √ 2 3 w √ w generates the sys tem v t = 1 √ 2 w w x , w t = v x + 1 √ 2 w w y . In tro ducing the v ariables V = v , W = √ 2 w , w e obtain V t = W x , W W t = V x + W y . Setting W = u t , V = u x w e arriv e at the dKP equation, u ty − u t u tt + u xx = 0 . F or the general sys tem (7), the in tegrabilit y conditions co ns titute a system of f ourth order PDEs for the Hamiltonian densit y h ( v , w ) , h w w ( h vv h w w − h 2 vw ) h vv vv = − 6 h 2 vw h 2 vv w + 3 h w w h vv h 2 vv w +4 h 2 vw h vw w h vv v − 2 h w w h vw h vv w h vv v + h 2 w w h 2 vv v , h w w ( h vv h w w − h 2 vw ) h vv vw = − 3 h 2 vw h vw w h vv w + 3 h w w h vw w h vv w h vv − 3 h w w h vw h 2 vv w + h 2 vw h vv v h w ww + h w w h vw h vw w h vv v + h 2 w w h vv w h vv v , h w w ( h vv h w w − h 2 vw ) h vv w w = − 2 h 2 vw h 2 vw w + 2 h vv h 2 vw w h w w − 3 h w w h vw h vw w h vv w + h w w h vv h w ww h w vv + h w w h vw h w ww h vv v + h 2 w w h vw w h vv v , h w w ( h vv h w w − h 2 vw ) h vw w w = − 2 h 2 vw h w ww h vw w − 3 h w w h 2 vw w h vw +3 h w w h vv h w ww h vw w + h w w h vw h w ww h vv w + h 2 w w h w ww h vv v , h w w ( h vv h w w − h 2 vw ) h w ww w = − 2 h 2 vw h 2 w ww − 2 h w w h w ww h vw h vw w − 3 h 2 w w h 2 vw w + 3 h w w h vv h 2 w ww + 4 h 2 w w h w ww h vv w . (16) 10 As an illustration of the metho d of hydrodynamic reductions, details of the deriv ation of these conditions are presen ted in the App endix. W e ha v e verifie d that system (16) is in in v olution, and its solution space is 10 -dim ensional. The integr abilit y conditions can b e represen ted in compact form as sd 4 h = d 3 hds + 3 h w w (2( dh w ) 2 − h w w d 2 h ) d et( dn ) where s = h 2 w w ( h vv h w w − h 2 vw ) , d k h denotes k -th s ymmetric differen tial of h , and n is the Hes s ian matrix of h . This system is in v arian t under an 11 -dimension al group of con tact s ymmetries whic h consists of the 9 -dimensional group of Lie-p oin t s ymmetries, v → av + b, w → pv + cw + d, h → αh + β v + γ w + δ, along with the t wo purely cont act infinitesimal generators, − Ω h v ∂ ∂ v − Ω h w ∂ ∂ w + (Ω − h v Ω h v − h w Ω h w ) ∂ ∂ h + (Ω v + h v Ω h ) ∂ ∂ h v + (Ω w + h w Ω h ) ∂ ∂ h w , with the generating f unction s Ω = h v h w and Ω = h 2 w , resp ectiv ely . T o solve this system w e apply the Legendre transformation, V = h v , W = h w , H = v h v + w h w − h, H V = v , H W = w . The transformed equations (7) tak e the Go duno v form (10), ( H V ) t = W x , ( H W ) t = V x + W y , and the in tegrabilit y conditions (16) simplify to H V V V V = 2 H 2 V V V H V V , H V V V W = 2 H V V W H V V V H V V , H V V W W = 2 H 2 V V W H V V , (17) H V W W W = 3 H V W W H V V W − H W W W H V V V H V V , H W W W W = 6 H 2 V W W − 4 H W W W H V V W H V V . This s y stem is also in inv olution, and can b e represen ted in compact form as S d 4 H = d 3 H dS − 6 H V V ( dW ) 2 det( dN ) , where S = H 2 V V , and N is the Hessian matrix of H . The system (17) is inv arian t under an 11-paramete r group of Lie-poin t symmetries, V → aV + bW + p, W → cW + d, (18) H → αH + β W 2 + γ V W + ǫV + κW + η . Th us, b oth cont act symmetry generator s of the sys tem (16) are mapp ed to p oin t symmetries of the sy s tem (17). All our classification results will b e perfo rmed mo dulo this equiv alence. 11 3.1 Classifica tion of in t egrable p otentia ls The main result of this section is a complete classification of solutions of the system (17): Theorem 2 Mo dulo the e quivalenc e gr oup (18), the ge neric inte gr able p otential H ( V , W ) is given by the formula H = V ln V σ ( W ) , (19) wher e σ is the W eierstr ass sigma-function: σ ′ /σ = ζ , ζ ′ = − ℘, ℘ ′ 2 = 4 ℘ 3 − g 3 . Its de gener ations c orr esp ond to H = V ln V W , (20) H = V ln V , (21) H = V 2 2 W + αW 7 , (22) as wel l as the fol lowing p olynomial p otentials: H = V 2 2 + V W 2 2 + W 4 4 , (23) H = V 2 2 + W 3 6 . (24) Pro of: In the analysis of Eqs . (17) it is conv enien t to consider t w o cases, H V V V 6 = 0 and H V V V = 0 . Case 1: H V V V 6 = 0 . Then Eqs. (17) 1 and (17) 2 imply H V V V = cH 2 V V , c = const , so that H V V = − 1 cV + f ( W ) . The s ubstitutio n in to (17) 3 implies that f ( W ) m ust b e linear. Mo dulo the equiv alence group w e can th us assume that H V V = 1 V , which gives H = V ln( V ) + g ( W ) V + h ( W ) . The substitution of this ansatz in to (17) 4 giv es h ′′′ = 0 , so that w e can set h equal to zero mo dulo the equiv alence group. Ultimately , the substitution into the last equation (17) 5 giv es g ′′′′ = 6 g ′′ 2 . The general solution of this equation is g = − ln σ ( W ) , where σ is the W eierstrass sigma- function defined as (ln σ ) ′ = ζ , ζ ′ = − ℘ with ℘ s atisfying ℘ ′ 2 = 4 ℘ 3 − g 3 (notice that g 2 = 0 ). Degenerations of this solution giv e g = − ln W and g = 0 . This leads to the three cases H = V ln V σ ( W ) , H = V ln V W , H = V ln V . Case 2: H V V V = 0 . Then H V V = a ( W ) , and (17) 3 implies a ′′ = 2 a ′ 2 a , 12 so that a ( W ) = 1 pW + q . There are again t w o sub cases to consider dep ending on whether H V V W is zero or non-zero. If H V V W 6 = 0 then p 6 = 0 and, mo dulo the equiv alence group, one can set H V V = 1 W . In tegrating tw ice with respect to V w e obtain H = V 2 2 W + b ( W ) V + c ( W ) . The substitution in to (17) 4 giv es b ′′′ = − 3 b ′′ /W . Mo dulo the equiv alence group, one can set b ( W ) = 1 /W . Th us w e get H = V 2 2 W + V W + c ( W ) . By a translation of V one can remo ve the in termediate term V W , lea ving H = V 2 2 W + c ( W ) . Then the substitution int o (17) 5 giv es c ′′′′ − 4 c ′′′ w = 0 . Mo dulo the equiv alence group, this giv es H = V 2 2 W + αW 7 . In the second sub case, H V V V = H V V W = 0 , one can normalize H as H = V 2 / 2 + a ( W ) V + b ( W ) . With this ansatz Eqs. (17) 1 -(17) 3 are trivial, while the last t w o equations imply , mo dulo the equiv alence group, that H = V 2 2 + αV W 2 + α 2 W 4 + β W 3 . The case α = 0 leads to H = V 2 2 + W 3 6 . The case α 6 = 0 leads, on appropriate rescalings, to H = V 2 2 + V W 2 2 + W 4 4 . This finishes the pro of of Theorem 2. 3.2 Dispersionless Lax pairs W e recall that a quasilinear system (1) is said to p ossess a disp ersionless Lax pair, S x = F ( u , S t ) , S y = G ( u , S t ) , (25) if it can b e reco vere d from the consistency condition S xy = S y x (w e p oin t out that the dep en- dence of F and G on S t is generally non-linear). Disp ersionless Lax pairs first app eared in the construction of the univ ersal Whitham hierarc h y , see [23 ] and references there in. It wa s observ ed in [39] that Lax pairs of this kind naturally arise from the usual ‘solitonic’ Lax pairs in the disp ersionless limit. It w as demonstrated in [14, 15] that, for a num b er of particula rly in terest- ing classes of systems, the existence of a disp ersionless Lax pair is e quivalent to the existence of h ydro dynamic reductions and, th us, to the in tegrabilit y . Lax pairs form a basis of the disp ersion- less ¯ ∂ -dressing metho d [5] and a no v el v ersion of the in v erse scattering transform ass o ciated with pararmeter-d ep enden t ve ctor fields [24]. I n this section w e calculate disp ersionless Lax pairs, S x = F ( V , W, S t ) , S y = G ( V , W, S t ) , 13 for all int egrable p oten tials app earing in Theorem 2. Th us, w e require that the consistency condition S xy = S y x results in the corresp onding equations (10). Without going into details of calculations, whic h are quite standard, we only state final results. Let us p oin t out that in all examples discussed below the first equation of the Lax pair do es not explicitly depend on V . The general class of Lax pairs of this k ind w as discussed in [28 ]. Case (24): H = V 2 2 + W 3 6 . The equations (10) tak e the form V t = W x , W W t = V x + W y . The corresp onding disp ersionless Lax pair is S x = S 2 t 2 − W, S y = − S 3 t 3 + S t W + V . Case (23): H = V 2 2 + V W 2 2 + W 4 4 . The equations (10) tak e the form V t + W W t = W x V W t + W V t + 3 W 2 W t = V x + W y . The corresp onding disp ersionless Lax pair is S x = S 4 t 4 − S t W , S y = − S 7 t 7 + W S 4 t + S t ( V − W 2 ) . Case (22): H = V 2 2 W , α = 0 . The equations (10) take the form V t W − V W t W 2 = W x , − V V t W 2 + V 2 W t W 3 = V x + W y . The corresp onding disp ersionless Lax pair is S x = − W 2 2 S 2 t , S y = V W S 2 t + W 5 5 S 5 t . Case (22): H = V 2 2 W + αW 7 . Without an y loss of generalit y w e will set α = 1 / 168 . The equations (10) tak e the form V t W − V W t W 2 = W x , − V V t W 2 + V 2 W t W 3 + 1 4 W 5 W t = V x + W y . 14 The corresp onding disp ersionless Lax pair is S x = − aW 2 2 , S y = aV W − aa ′ W 5 10 , here a = a ( S t ) s atisfies the ODE 3 a ′ 2 − 5 = 2 aa ′′ . Case (21): H = V log V . The equations (10) take the form V t = V W x , 0 = V x + W y . The corresp onding disp ersionless Lax pair is S x = − log( W + S t ) , S y = V W + S t . Case (20): H = V log V W . The equations (10) tak e the form V t V − W t W = W x , V W t W 2 − V t W = V x + W y . The corresp onding disp ersionless Lax pair is S x = − ln( S t + W ) − ǫ ln( S t + ǫW ) − ǫ ln( S t + ǫ 2 W ) , ǫ = e 2 πi 3 , S y = − 3 V W S t W 3 + S 3 t . Case (19): H = V log V σ ( W ) . W e recall that (log σ ) ′ = ζ and ζ ′ = − ℘ where ℘ = ℘ ( w, 0 , g 3 ) is the W eierstrass ℘ -function, ℘ ′ 2 = 4 ℘ 3 − g 3 . The equations (10) take the form 1 V V t − ζ ( W ) W t = W x , − ζ ( W ) V t + V ℘ ( W ) W t = V x + W y . The corresp onding disp ersionless Lax pair is of the form S x = F ( W , S t ) , S y = V A ( W , S t ) , where the functions F and A satisfy the equations F W = − A, F ξ = A W A − ζ ( W ) and A ξ = ℘ ( W ) − A 2 W A 2 , A W W = 2 A 2 W A − 2 A℘ ( W ) , 15 resp ectiv ely (here ξ = S t ). Setting A = 1 /u one can rewrite the last t wo equations for A in the equiv alen t form, u ξ = u 2 W − ℘ ( W ) u 2 , u W W = 2 ℘ ( W ) u. (26) Here the second equation is a particular case of the Lame equation. It has tw o linearly indepen- den t solutions [3], e − W ζ ( α ) σ ( W + α ) σ ( W ) σ ( α ) , e W ζ ( α ) σ ( W − α ) σ ( W ) σ ( α ) , where α is the zero of the ℘ -function: ℘ ( α ) = 0 . Th us, one can set u = − e − W ζ ( α ) σ ( W + α ) σ ( W ) σ ( α ) a ( ξ ) + e W ζ ( α ) σ ( W − α ) σ ( W ) σ ( α ) b ( ξ ) . The substitution int o ( 26 ) 1 implies a pair of ODEs for a ( ξ ) and b ( ξ ) , a ′ = ℘ ′ ( α ) b 2 , b ′ = − ℘ ′ ( α ) a 2 . This s y stem can b e s olved in the form a ( ξ ) = c τ ℘ ′ ( ξ , 0 , g 3 ) − 1 τ ℘ ′ ( ξ , 0 , g 3 ) + 1 , b ( ξ ) = c γ ℘ ( ξ , 0 , g 3 ) τ ℘ ′ ( ξ , 0 , g 3 ) + 1 , where the constan ts c, τ and γ are defined as τ = 1 √ 3 g − 1 / 2 3 , γ = 2 g − 1 / 3 3 , c = √ 3 ℘ ′ ( α ) g 1 / 6 3 . Setting g 3 = 1 , ℘ ′ ( α ) = i one obtains a ( ξ ) = − i √ 3 ℘ ′ ( ξ , 0 , 1) − √ 3 ℘ ′ ( ξ , 0 , 1) + √ 3 , b ( ξ ) = − i 6 ℘ ( ξ , 0 , 1) ℘ ′ ( ξ , 0 , 1) + √ 3 . Ultimately , u = i √ 3 e − W ζ ( α ) σ ( W + α ) σ ( W ) σ ( α ) ℘ ′ ( ξ ) − √ 3 ℘ ′ ( ξ ) + √ 3 − 6 i e W ζ ( α ) σ ( W − α ) σ ( W ) σ ( α ) ℘ ( ξ ) ℘ ′ ( ξ ) + √ 3 . The functions A and F can b e reconstructed via A = 1 /u , F W = − 1 /u, F ξ = − u W u − ζ ( W ) . 4 Hamiltoni an systems of t yp e (8) In this section w e consider Hamiltonian systems of the form (8),  v w  t = "  2 v w w 0  d dx +  0 v v 2 w  d dy +  v x v y w x w y  #  h v h w  , or, explicitly , v t = (2 v h v + w h w − h ) x + ( v h w ) y , w t = ( w h v ) x + (2 w h w + v h v − h ) y . 16 The integr abilit y conditions constitute a system of fourth order PDEs for the H amiltonian densit y h ( v , w ) whic h is not presen ted here due to its complexit y . W e hav e v erified that this system is in in volutio n, and its solution space is 10 -dim ensional. It is in v arian t under an 8 -dimensional group of Lie-poin t symmetries, v → av + bw , w → cv + dw , h → αh + β v + γ w + δ, along with the t wo purely cont act infinitesimal generators, − Ω h v ∂ ∂ v − Ω h w ∂ ∂ w + (Ω − h v Ω h v − h w Ω h w ) ∂ ∂ h + (Ω v + h v Ω h ) ∂ ∂ h v + (Ω w + h w Ω h ) ∂ ∂ h w , with the generating functions Ω = h v ( v h v + w h w ) and Ω = h w ( v h v + w h w ) , resp ectiv ely . As in the previous t w o cases, the in tegrabilit y conditions for the Hamiltonian densit y h ( v , w ) simplify under the Legendre transformation, V = h v , W = h w , H = v h v + w h w − h, H V = v , H W = w , whic h brings the corresponding equations (8) in to the Go duno v form (11), ( H V ) t = ( V H V + H ) x + ( W H V ) y , ( H W ) t = ( V H W ) x + ( W H W + H ) y . Remark. Equations (11) coincide with the s o-called EPDiff equations [22], m t − u × curl m + ∇ ( u , m ) + m div u = 0 , where u = − ( V , W, 0) and m = ( H V , H W , 0) . These equations hav e a geometric interpr etation as the Euler-P oincar ´ e equations for the geo desic motion on the group of diffeomorphisms. 17 The in tegrabilit y conditions for the system (11) take the form S H V V V V =2( H W H V V V − H V H V V W ) 2 + 4 H 2 V ( H 2 V V W − H V V V H V W W ) + 12 H W H 2 V V H V V W − 12 H W H V V H V W H V V V − 24 H V H V V H V W H V V W + 8 H V H 2 V W H V V V + 6 H V H 2 V V H V W W + 10 H V H V V H W W H V V V + 12 H 2 V V ( H 2 V W − H V V H W W ) , S H V V V W = H V V V (2 H 2 W H V V W − 4 H V H W H V W W − H 2 V H W W W ) + 3 H 2 V H V V W H V W W − 3 H W H V V H V W H V V W − 4 H W H 2 V W H V V V − 6 H V H 2 V W H V V W + 15 2 H W H 2 V V H V W W − 6 H V H V V H V W H V W W + 3 2 H V H 2 V V H W W W − 1 2 H W H V V H W W H V V V + 3 2 H V H V V H W W H V V W + 9 H V H V W H W W H V V V + 12 H V V H V W ( H 2 V W − H V V H W W ) , S H V V W W =2( H V H V W W − H W H V V W ) 2 + 2 H V H W ( H V V W H V W W − H V V V H W W W ) − 8 H 2 V W ( H W H V V W + H V H V W W ) + 6 H W H V V H V W H W W V + 6 H V H W W H V W H V V W − H V V H W W ( H W H V V W + H V H V W W ) + 3 H W H 2 V V H W W W + 3 H V H 2 W W H V V V + 4(2 H 2 V W + H V V H W W )( H 2 V W − H V V H W W ) , S H V W W W = H W W W (2 H 2 V H V W W − 4 H V H W H V V W − H 2 W H V V V ) + 3 H 2 W H V V W H V W W − 3 H V H W W H V W H V W W − 4 H V H 2 V W H W W W − 6 H W H 2 V W H V W W + 15 2 H V H 2 W W H V V W − 6 H W H W W H V W H V V W + 3 2 H W H 2 W W H V V V − 1 2 H V H V V H W W H W W W + 3 2 H W H V V H W W H V W W + 9 H W H V W H V V H W W W + 12 H W W H V W ( H 2 V W − H V V H W W ) , S H W W W W =2( H V H W W W − H W H V W W ) 2 + 4 H 2 W ( H 2 V W W − H W W W H V V W ) + 12 H V H 2 W W H V W W − 12 H V H W W H V W H W W W − 24 H W H W W H V W H V W W + 8 H W H 2 V W H W W W + 6 H W H 2 W W H V V W + 10 H W H W W H V V H W W W + 12 H 2 W W ( H 2 V W − H V V H W W ) , where S = H 2 W H V V − 2 H V H W H V W + H 2 V H W W . This system is manifestly s y mmetric under the in terc hange of V and W , and can be represen ted in compact form as S d 4 H = 2 d 3 H dS − 6( dH ) 2 det( dN ) + 6 dH d 2 H d (det N )+ 12( H V dH W − H W dH V )( d 2 H V dH W − d 2 H W dH V ) − 12 det N ( d 2 H ) 2 , (27) 18 where N is the Hes sian matrix of H . W e ha v e verifi ed that the system (27) is in in volutio n, and is in v arian t under a 10 -dimensional group of Lie-point symmetries generated b y pr o jectiv e transformations of V and W along with affine transformations of H , V → aV + bW + c pV + q W + r , W → αV + β W + γ pV + q W + r , H → µH + ν. The corresp onding infinitesimal generators are: 2 translations : ∂ ∂ V , ∂ ∂ W ; 4 linear transf orm atio ns : V ∂ ∂ V , W ∂ ∂ V , V ∂ ∂ W , W ∂ ∂ W ; 2 p r o jectiv e transformations : V 2 ∂ ∂ V + V W ∂ ∂ W , V W ∂ ∂ V + W 2 ∂ ∂ W ; 2 affine transf ormations of H : ∂ ∂ H , H ∂ ∂ H . (28) Pro jective transformations (28 ) constitute the group of ‘canonical’ transformations of equations (11): combi ned with appropriate linear c hanges of the indep enden t v ariables x, y , t , they lea ve equations (11) form-in v arian t. Let us consider, for instance, the pro jective transformation ˜ V = 1 /V , ˜ W = W /V . A direct calculation sho ws that the transformed equations tak e the form ( ˜ V H ˜ V + H ) t = ( H ˜ V ) x + ( ˜ W H ˜ V ) y , ( ˜ V H ˜ W ) t = ( H ˜ W ) x + ( ˜ W H ˜ W + H ) y . They assume the origi nal form (11) on the iden tification t → x, x → t, y → − y . In other w ords, the pro jective in v ariance of the in tegrabilit y conditions is a manifestation of the in v ariance of the Hamiltonian formalism (8) under arbitrary linear transformations of the independen t v ariables. This is analogous to the well-kno wn in v ariance of Hamiltonian s tructur es of h ydro dynamic t yp e in 1 +1 dimensions under linear transformations of x and t [34]. W e emphasise that this inv ariance is not presen t in the case of constant co efficien t Poisson brac k ets. 4.1 Classifica tion of in t egrable p otentia ls The analysis of this section is somewhat similar to the classification of in tegrable Lagrangians of the form R u t g ( u x , u y ) dxdy dt prop osed in [17]. This suggests that there ma y b e a closer link b et w een these t w o classes of equations. The main result of this section is the follo wing Theorem 3 T he generic inte gr able p otential of typ e (11) is given by the series H = 1 W g ( V )  1 + 1 g 1 ( V ) W 6 + 1 g 2 ( V ) W 12 + ...  . Her e g ( V ) satisfies the fourth or der ODE, g ′′′′ (2 gg ′ 2 − g 2 g ′′ ) + 2 g 2 g ′′′ 2 − 20 g g ′ g ′′ g ′′′ + 16 g ′ 3 g ′′′ + 18 g g ′′ 3 − 18 g ′ 2 g ′′ 2 = 0 , whose gener al s olution c an b e r epr esente d in p ar ametric for m as g = w 2 ( t ) , V = w 1 ( t ) w 2 ( t ) , 19 wher e w 1 ( t ) and w 2 ( t ) ar e two line arly indep endent s olutions of the hyp er ge ometric e quation t (1 − t ) d 2 w/dt 2 − 2 9 w = 0 . The c o efficients g i ( V ) ar e c ertain explicit expr essions in terms of g ( V ) , e.g., g 1 ( V ) = gg ′′ − 2 g ′ 2 and so on. De gener ations of this solution c orr esp ond to H = 1 W g ( V ) , H = 1 W V , H = V − W log W, H = V − W 2 / 2 . In the rest of this section we pro vide details of the classification, and discuss v arious prop erties and represen tations of the generic solution. The system (27 ) for H ( V , W ) is not straigh tforw ard to solv e explicitly . W e will s tart with the inv estigation of sp ecial solutions whic h are in v arian t under v arious one-parameter subgroups of the equiv alence group. In this case the system of in tegrabilit y conditions f or H ( V , W ) reduces to ODEs whic h are easier to solv e. Up to conjugation and normalisation, there exist four essentia lly differen t one-parameter subgroups of the pro jectiv e group S L (3) , with the infinitesimal generators αV ∂ ∂ V + W ∂ ∂ W , V ∂ ∂ V + ∂ ∂ W , W ∂ ∂ V + ∂ ∂ W , ∂ ∂ V . Com bined with the operators ∂ /∂ H , H ∂ /∂ H this leads to the follo wing list of eleve n essen tially differen t ‘ansatzes’ go ve rning in v arian t solutions (in what follo ws w e do not consider degenerate solutions for whic h the expression S = H 2 W H V V − 2 H V H W H V W + H 2 V H W W equals zero): Case 1. Solutions inv arian t under the op erator ∂ /∂ V + ∂ /∂ H are described b y the ansatz H = V + F ( W ) . The in tegrabilit y conditions imply F ′′ F ′′′′ − 2 F ′′′ 2 = 0 . Mo dulo the equiv alence transformations this leads to int egrable potenti als H = V − W 2 / 2 and H = V − W log W , whic h constitute the last tw o cases of Theorem 3. Applying to the first p oten tial transformations f rom the equiv alence group w e obtain in tegrable p oten tials of the form H ( V , W ) = Q ( V , W ) l 2 ( V , W ) where Q and l are quadratic and linear f orms, resp ective ly (not necessarily homogeneous). The in tegrabilit y implies that the line l = 0 is tangen tial to the conic Q = 0 on the V , W plane. An y suc h p oten tial can b e reduced to the form H = V − W 2 / 2 by a pro jective transformation which sends l to the line at infinit y . Case 2. Solutions in v arian t under the op erator ∂ /∂ W + H ∂ /∂ H are describ ed b y the ansatz H = e W F ( V ) . This case giv es no non-trivial solutions. Case 3. Solutions in v arian t under the op erator W ∂ /∂ V + ∂ /∂ W are describ ed b y the ansatz H = F ( V − W 2 / 2) . A simple analysis leads to the only p olynomial p oten tial H = V − W 2 / 2 , the same as in Case 1. Case 4. Solutions in v arian t under the op erator W ∂ /∂ V + ∂ /∂ W + ∂ /∂ H are describ ed b y the ansatz H = W + F ( V − W 2 / 2) . In this case F turns out to b e linear s o that, modulo translations in W , we again arriv e at the same p oten tial as in Case 1. Case 5. Solutions inv arian t under the op erator W ∂ /∂ V + ∂ /∂ W + H ∂ /∂ H are describ ed b y the ansatz H = e W F ( V − W 2 / 2) . A detailed analysis sho ws that this case gives no non-trivial solutions. Case 6. Solutions in v arian t under the op erator V ∂ /∂ V + ∂ /∂ W are describ ed by the ansatz H = F ( W − ln V ) . This case giv es no non-trivial solutions. Case 7. Solutions inv arian t under the operator V ∂ /∂ V + ∂ /∂ W + ∂ /∂ H are describ ed b y the ansatz H = W + F ( W − ln V ) . This case giv es no non-trivial s olution s. 20 Case 8. Solutions in v arian t under the op erator V ∂ /∂ V + ∂ /∂ W + µH ∂ /∂ H are describ ed by the ansatz H = e µW F ( W − ln V ) , also no non-trivial solutions. Case 9. Solutions in v arian t under the op erator αV ∂ /∂ V + W ∂ /∂ W are describ ed b y the ansatz H = F ( W α /V ) . Here one can ass ume α 6 = 0 , 1 . A s traight forw ard substitution implies that, without an y loss of generalit y , one can assume F to b e linear, while the parameter a can only tak e t wo v alues: a = 2 or a = − 1 . This results in the tw o rational p oten tials H = W 2 /V and H = 1 /V W . Notice that they are related b y the pro jectiv e transforma tion W → 1 /W, V → V /W . Case 1 0. Solutions in v arian t under the op erator αV ∂ /∂ V + W ∂ /∂ W + ∂ /∂ H are describ ed b y the ansatz H = ln W + F ( W α /V ) . This case giv es no non-trivial solutions. Case 11. Solutions in v arian t under the op erator αV ∂ /∂ V + W ∂ /∂ W + µH ∂ /∂ H are described b y the ansatz H = W µ F ( W α /V ) . A detailed analysis sho ws that, mo dulo the equiv alence group and the solutio ns alre ady discussed ab o ve, the only essen tially new p ossibilit y corresp onds to the ansatz H = 1 W g ( V ) (here µ = − 1 , α = 0 ). It leads to the fourth order ODE for g = g ( z ) , g ′′′′ (2 gg ′ 2 − g 2 g ′′ ) + 2 g 2 g ′′′ 2 − 20 g g ′ g ′′ g ′′′ + 16 g ′ 3 g ′′′ + 18 g g ′′ 3 − 18 g ′ 2 g ′′ 2 = 0 , (29) whic h p ossesses a remark able S L (2 , R ) -in v ariance inherited from (28 ): ˜ z = αz + β γ z + δ , ˜ g = ( γ z + δ ) g ; (30) here α, β , γ , δ are arbitrary constan ts s uc h that αδ − β γ = 1 . Moreo v er, there is an ob v ious scaling symmetry g → λg . The equation (29) can be linearised as f ollows. In tro ducing h = g ′ /g , whic h means factoring out the scaling symmetry , w e first rewrite it in the form h ′′′ ( h ′ − h 2 ) − 2 h ′′ 2 + 12 hh ′ h ′′ − 4 h 3 h ′′ − 15 h ′ 3 + 9 h 2 h ′ 2 − 3 h 4 h ′ + h 6 = 0; (31) the corresp onding symmetry group mo difies to ˜ z = αz + β γ z + δ , ˜ h = ( γ z + δ ) 2 h + γ ( γ z + δ ) . (32) W e p oin t out that the same symmetry o ccurs in the case of the Chazy equation, s ee [1], p. 342. The presence of the S L (2 , R ) -symmetry of this t yp e implies the linearisabilit y of the equation under s tudy . One can formu late the follo wing general statemen t whic h is , in fact, con tained in [9]. Prop osition 1. A ny thir d or der ODE of the f orm F ( z , h, h ′ , h ′′ , h ′′′ ) = 0 , whi ch is invariant under the action of S L (2 , R ) as sp e cifie d by (32), c an b e line arise d by a substitution h = d dz ln w 2 , z = w 1 w 2 (33) wher e w 1 ( t ) and w 2 ( t ) ar e two line arly indep endent solutions of a li ne ar e quation d 2 w/dt 2 = V ( t ) w with the W r onskian W normalise d as W = w 2 dw 1 /dt − w 1 dw 2 /dt = 1 . The p otential V ( t ) dep ends on the given thir d or der ODE, and c an b e efficiently r e c onstructe d. In p articular, the gener al solution of the e quation (31) i s given by p ar ametric formulae (33) wher e w 1 ( t ) and w 2 ( t ) ar e two line arly indep endent s olutions of the hyp er ge ometric e quation d 2 w/dt 2 = 2 9 1 t (1 − t ) w with W = 1 . 21 Pro of: Our presen tation follows [17]. Let us consider a linear ODE d 2 w/dt 2 = V ( t ) w , tak e t w o linearly independent solutions w 1 ( t ) , w 2 ( t ) with the W ronskian W = 1 , and in tro duce new dep enden t and indep enden t v ariables h, z by parametric relations h = d dz ln w 2 , z = w 1 w 2 . Using the form ulae dt/dz = w 2 2 and h = w 2 dw 2 /dt , one obtains the iden tities h ′ − h 2 = w 4 2 V , h ′′ − 6 hh ′ + 4 h 3 = w 6 2 dV /dt, h ′′′ − 12 hh ′′ − 6( h ′ ) 2 + 48 h 2 h ′ − 24 h 4 = w 8 2 d 2 V /dt 2 , where prime denotes differen tiation with resp ect to z . Th us, one arriv es at the relation s I 1 = ( h ′′ − 6 hh ′ + 4 h 3 ) 2 ( h ′ − h 2 ) 3 = ( dV /dt ) 2 V 3 , I 2 = h ′′′ − 12 hh ′′ − 6( h ′ ) 2 + 48 h 2 h ′ − 24 h 4 ( h ′ − h 2 ) 2 = d 2 V /dt 2 V 2 . W e p oin t out that I 1 and I 2 are the simplest second and third order differen tial in v arian ts of the action (32 ) whose infinitesimal generators, prolonged to the third order jets z , h, h ′ , h ′′ , h ′′′ , are of the form X 1 = ∂ z , X 2 = z ∂ z − h∂ h − 2 h ′ ∂ h ′ − 3 h ′′ ∂ h ′′ − 4 h ′′′ ∂ h ′′′ , X 3 = z 2 ∂ z − (2 z h + 1) ∂ h − (2 h + 4 z h ′ ) ∂ h ′ − (6 h ′ + 6 z h ′′ ) ∂ h ′′ − (12 h ′′ + 8 z h ′′′ ) ∂ h ′′′ ; notice the standard com m utation relations [ X 1 , X 2 ] = X 1 , [ X 1 , X 3 ] = 2 X 2 , [ X 2 , X 3 ] = X 3 . One can verify that the Lie deriv atives of I 1 , I 2 with resp ect to X 1 , X 2 , X 3 are inde ed zero. Thus, an y third order ODE whic h is inv arian t under the S L (2 , R ) -action (32 ), can b e represen ted in the form I 2 = F ( I 1 ) where F is an arbitrary function of one v ariable. The corresp onding p oten tial V ( t ) has to satisfy the equation d 2 V /dt 2 V 2 = F  ( dV /dt ) 2 V 3  . This simple schem e pro duces many the w ell-kno wn equations, f or instance, the relation I 2 = − 24 implies the Chazy equation for h , that is, h ′′′ − 12 hh ′′ + 18( h ′ ) 2 = 0 . The corresp onding p oten tial satisfies the equation d 2 V /dt 2 = − 24 V 2 . Similarly , the choice I 2 = I 1 − 8 results in the ODE h ′′′ = 4 hh ′′ − 2( h ′ ) 2 + ( h ′′ − 2 hh ′ ) 2 h ′ − h 2 whic h, under the substitution h = y / 2 , coincides with the equatio n (4.7) from [2]. The p oten tial V satisfies the equation V d 2 V /dt 2 = ( dV /dt ) 2 − 8 V 3 . The relation I 2 = I 1 − 9 giv es h ′′′ ( h ′ − h 2 ) = h 6 − 3 h 4 h ′ + 9 h 2 ( h ′ ) 2 − 3( h ′ ) 3 − 4 h 3 h ′′ + ( h ′′ ) 2 . This equation app eared in [17] in the con text of first order in tegrable Lagran gians. The corresp onding p oten tial V satisfies the equation V d 2 V /dt 2 = ( dV /dt ) 2 − 9 V 3 . Finally , the relation I 2 = 2 I 1 + 9 coincides with the equation (31). The corresp onding p oten tial V satisfies the equation V d 2 V /dt 2 = 2( dV /dt ) 2 + 9 V 3 . It remains to p oin t out that 22 the general solution to the last equation for V is given by V = − 2 9 1 t 2 + at + b . Without an y loss of generalit y one can set V = 2 9 1 t (1 − t ) . It this case the linear equation d 2 w/dt 2 = V ( t ) w takes the hypergeometric form corresp onding to the parameter v alues a = − 1 3 , b = − 2 3 , c = 0 : t (1 − t ) d 2 w /dt 2 − 2 9 w = 0 . This finishes the pro of of Proposition 1. As h = g ′ /g , this immediately implies the f ollo wing form ula for the general solution of (29 ): Prop osition 2. The gener al solution of the e quation (29) is given by p ar ametric f ormulae g = w 2 , z = w 1 w 2 , wher e w 1 and w 2 ar e two line arly indep endent solutions to the hyp er ge ometric e quation t (1 − t ) d 2 w/dt 2 − 2 9 w = 0 . Remark. Expansions at zero giv e w 1 = w 2 ln t + 9 2 + t 2 3 + 211 t 3 1458 + ..., w 2 = t + t 2 9 + 10 t 3 243 + ..., so that g = w 2 = t + t 2 9 + 10 t 3 243 + ..., z = w 1 w 2 = ln t + 9 2 t − 1 2 + 11 t 54 + 34 t 2 729 + 715 t 3 39366 + .... Solving the first relation for t in terms of g and substituting in to the second, one gets an implicit relation connecting g and z , z = ln g − 9 g + 1 2 + 112 g 27 + 289 g 2 486 + 4381 g 3 39366 + .... Similarly , expansions at infinit y give w 1 = t 2 / 3  1 − 1 3 t − 2 45 t 2 − ...  , w 2 = t 1 / 3  1 − 1 6 t − 5 126 t 2 − ...  , z = w 1 w 2 = t 1 / 3  1 − 1 6 t − 41 1260 t 2 − ...  , so that g = w 2 can b e represen ted explicitly as g ( z ) = z  1 − a z 6 − 1165 143 a 2 z 12 − 52800 35 46189 a 3 z 18 − ...  , here a = 1 / 140 . F or other v alues of a this form ula represen ts the general solution to (29) in the form g = z (1 − a/z 6 − b/z 12 − c/z 18 − ... ) . The corresp onding p oten tial H ( V , W ) takes the form H ( V , W ) = 1 W g ( V ) = 1 V W  1 + a V 6 + 1308 143 a 2 V 12 + ...  . It can be view ed as a perturbation of the p oten tial H = 1 V W constructed b efore. Compute r exp erimen ts sho w that the ‘generic’ in tegrable p oten tial H ( V , W ) can b e represen ted as H = X j,k ≥ 0 a j k V 6 j +1 W 6 k + 1 = 1 V W  1 + 1 V 6 + 1 W 6 − 24 V 6 W 6 + 1308 143 V 12 + 1308 143 W 12 + . . .  . 23 An alternativ e represen tation of this solution, H = 1 W g ( V )  1 + 1 g 1 ( V ) W 6 + 1 g 2 ( V ) W 12 + ...  , can b e obtained b e rearrangin g terms in the ab o v e sum. The substitution of this expression into the in tegrabilit y conditions implies that g ( V ) has to s atisfy the fourth order ODE (29), g 1 ( V ) is expressed in terms of g ( V ) via the Rankin-Cohen-t yp e op eration, g 1 ( V ) = g g ′′ − 2 g ′ 2 (recall that, due to (30), g ( V ) transforms as a mo dular form of w eigh t 1), and so on. This finishes the pro of of Theorem 3. In Sect. 5 w e pro vide a parametrisation of the generic in tegrable p oten tial b y generalised h yp ergeometric functions. 4.2 Dispersionless Lax pairs Here we presen t Lax pairs for some of the s implest p oten tials found in the previous s ection. Case H = V − W 2 / 2 . The corresponding sys tem (11) tak es the form W y = W W x − 2 V x , W t = V W x − 5 W V x + 3 W 2 W x − V y , and p ossesses the Lax pair S y = − W S x + 1 S 2 x , S t = ( V − W 2 ) S x + W S 2 x − 2 5 S 5 x . Case H = V − W log W . The corresp onding system (11) takes the form W y = ( W ln W − 2 V ) x , W t /W = ( V ln W + V ) x + (2 W ln W + W − V ) y , and p ossesses the Lax pair S y = − S x (ln W + 1) + p ( S x ) , S t = S x ( V − W ln W − W ) − 1 2 S x p ′ ( S x ) W where p ( z ) satisfies the ODE z p ′′ + p ′ 2 + 3 p ′ = 0 . This giv es p ′ = 3 z 3 − 1 so that p ( S x ) = ln( S x − 1) + ǫ ln( S x − ǫ ) + ǫ 2 ln( S x − ǫ 2 ) , ǫ = e 2 πi 3 . (34) Case H = 1 W V . The corresp onding s ystem (11) tak es the form  1 V 2 W  t =  1 V 2  y ,  1 V W 2  t =  1 W 2  x , (35) and p ossesses the Lax pair S x = a ( S t ) /V , S y = b ( S t ) /W where the functions a ( z ) and b ( z ) , z = S t , satisfy a pair of ODEs a ′ = 1 − a 2 b , b ′ = 1 − b 2 a . These equations can b e s olv ed in paramet ric form as a ( z ) = ℘ ′ ( p ) + λ 2 ℘ ( p ) , b ( z ) = ℘ ′ ( p ) − λ 2 ℘ ( p ) , z = − 2 ζ ( p ) , 24 where ℘ ( p ) and ζ ( p ) are the W eierstrass functions, ℘ ′ 2 = 4 ℘ 3 + λ 2 , ζ ′ = − ℘ . The resulting Lax pair can b e written in parametric form as S x = 1 V ℘ ′ ( p ) + λ 2 ℘ ( p ) , S y = 1 W ℘ ′ ( p ) − λ 2 ℘ ( p ) , S t = − 2 ζ ( p ) , or, equiv alen tly , S x S y ( V S x − W S y ) = λ V W , S x S y = ℘ ( p ) V W , S t = − 2 ζ ( p ) . Notice that equations (35) coincide with the Euler-L agrange equations corresp onding to the Lagrangian densit y R u x u y u 2 t dxdy dt up on setting V = u t 2 u x , W = u t 2 u y (w e thank Maxim P a vlov for p oin ting this out). In this context, the ab o ve Lax pair app eared previously in [33]. 5 Go duno v systems and g eneralized h yp ergeometric functions In this section w e dev elop the general theory of Go duno v’s s ystems [21], and describ e the Go- duno v form of n -comp onen t quasilinear systems constructed in [27, 29] in terms of generalized h yp ergeometric functions. In particular, this pro v ides a Go duno v represen tation for any generic 2-component integra ble s y stem [14, 27]. Applied to the system (11), this construction gives a parametrisation of the generic in tegrable p oten tial H ( V , W ) in the form (13). In Sect. 5.1 we recall the main asp ects of the Go duno v representa tion, and clarify its sy m- metry prop erties. In Sect. 5.2 we construct the Go duno v form for quasilinear sys tems f ound in [27, 29 ] in terms of generalised hypergeometric functions. In Sect. 5.3 w e sp ecialise this construction to in tegrable p oten tials H ( V , W ) of the t yp e (11), th us pro ving Theorem 4. 5.1 Go duno v systems A 2 + 1 dimensional quasilinear system in n unkno wns v = ( v 1 , ..., v n ) is said to p ossess a Go duno v represen tation [21 ] if it can b e written in the conserv ative form, ( F 0 ,i ) t + ( F 1 ,i ) x + ( F 2 ,i ) y = 0 , i = 1 , . . . , n, (36) where p oten tials F 0 , F 1 , F 2 are functions of v , and F α,j = ∂ F α /∂ v j , α = 0 , 1 , 2 . Any suc h system automaticall y p ossess es an extra conserv ation la w, L ( F 0 ) t + L ( F 1 ) x + L ( F 2 ) y = 0 , (37) where L denotes the Legendre transform: L ( F α ) = F α,k v k − F α . Many systems of ph ysical origin are known to b e represen table in the Go duno v form. This represen tation is widely used for analytical/n umerical treatmen t of quasilinear sys tems. Note that the systems (9)-(11) are written in the Go duno v form. F or example, in the case (11) w e hav e n = 2 , v 1 = V , v 2 = W , F 0 = − H , F 1 = V H , F 2 = W H . R ecall that a 2 + 1 dimensional quasilinear system of n equations for n unkno wns p osses s es a Go duno v form iff it p ossesses n + 1 conserv ation laws of h ydro dynamic type. The follo wing fact will b e useful: Prop osition 1. The Go dunov r epr esentation (36) is form-invariant under the pr oje ctive action of GL n +1 define d as ˜ v i = l i ( v ) l ( v ) , ˜ F α = F α l ( v ) , 25 her e l i , l ar e li ne ar (inhomo gene ous) f orms in v . Pro of: With an y Go duno v sy stem w e asso ciate the f ollo wing geometric ob jects. Let us consider an auxiliary ( n + 1) -dimen sional affine space A n +1 with co ordinates x 1 , . . . , x n , x n +1 . With any p oten tial F α w e asso ciate an n -parameter family of h yp erplanes, x n +1 − x k v k + F α ( v ) = 0 . (38) The env elop e of this family is a h yp ersurface M n F α ⊂ A n +1 defined parametrically as ( x 1 , . . . , x n , x n +1 ) = ( F α, 1 , . . . , F α,n , L ( F α )) . Notice tha t componen ts of its p osition vector are the conserv ed densities app earing in Eqs. (36), (37). By construction, hypersurfaces M n F 0 , M n F 1 and M n F 2 ha ve parallel tangen t hyperplanes at the poin ts corresponding to the same v alues of the parameters v i . Let us now apply an arbit rary affine transformatio n in A n +1 , ˜ x = Ax , where A is a constan t ( n + 1) × ( n + 1) matrix. This will transform Eq. (38) to ˜ x n +1 − ˜ x k ˜ v k + ˜ F α ( ˜ v ) = 0 (39) where the transformation v → ˜ v will automatically b e pro jectiv e: ˜ v i = l i ( v ) l ( v ) , ˜ F α = F α l ( v ) ; here l i , l are linear (inhomog eneous) f orms in v . Since affine transformations preserv e the prop erties of b eing parallel/tangen tial, they naturally act on the class of Go duno v’s systems. 5.2 Go duno v form of integrable quasilin ear systems and generalized h yp er- geometric functions Let H s 1 ,...,s n +2 b e the space of s olutions of the system ∂ 2 h ∂ u i ∂ u j = s i u i − u j · ∂ h ∂ u j + s j u j − u i · ∂ h ∂ u i , i, j = 1 , ..., n, i 6 = j, (40) and ∂ 2 h ∂ u i ∂ u i = −  1 + n +2 X j =1 s j  s i u i ( u i − 1) h + s i u i ( u i − 1) n X j 6 = i u j ( u j − 1) u j − u i · ∂ h ∂ u j +  n X j 6 = i s j u i − u j + s i + s n +1 u i + s i + s n +2 u i − 1  ∂ h ∂ u i , (41) for one unkno wn function h ( u 1 , . . . , u n ) . Here s 1 , ..., s n +2 are arbitrary constan ts . Elemen ts of H s 1 ,...,s n +2 are examples of the so-called generalised hypergeometric functions, see [18], [29]. It is know n that dim H s 1 ,...,s n +2 = n + 1 . Let g 0 , g 1 , ..., g n b e a basis of H s 1 ,...,s n +2 . Choose a time t q for each elemen t g q of this basis. Here q runs from 0 to n . It is known [27], [29] that for eac h pairwise distinct q , r, s running from 0 to n the system X 1 ≤ i ≤ n,i 6 = j ( g q ,u j g r,u i − g r,u j g q ,u i ) u j ( u j − 1) u i,t s − u i ( u i − 1) u j,t s u j − u i + σ · ( g q g r,u j − g r g q ,u j ) u j,t s + 26 X 1 ≤ i ≤ n,i 6 = j ( g r,u j g s,u i − g s,u j g r,u i ) u j ( u j − 1) u i,t q − u i ( u i − 1) u j,t q u j − u i + σ · ( g r g s,u j − g s g r,u j ) u j,t q + (42) X 1 ≤ i ≤ n,i 6 = j ( g s,u j g q ,u i − g q ,u j g s,u i ) u j ( u j − 1) u i,t r − u i ( u i − 1) u j,t r u j − u i + σ · ( g s g q ,u j − g q g s,u j ) u j,t r = 0 , where j = 1 , ..., n, σ = 1 + s 1 + ... + s n +2 , p ossesses a disp erssionless Lax represen tation and an infinit y of hydrodynamic reductions. Moreov er, any generic int egrable 3-dimensional hydrody- namic type s ystem with tw o unkno wns is isomorphic to a system of the form (42), see [27]. It is also kno wn that the sys tem (42) p ossess es n + 1 conserv ation la ws of h ydro dynamic t y pe, see [14] for n = 2 and [29 ] for general n . Therefore, it p ossesses a Go duno v represent ation whic h can b e constructed explicitly in the follo wing wa y . Theorem 5 L et h 0 , h 1 , ..., h n b e a b asis of H 2 s 1 ,..., 2 s n +2 . F or e ach α 6 = β = 0 , 1 , ..., n let f α,β b e a solution of the i nhomo gene ous line ar system ∂ 2 h ∂ u i ∂ u j = 2 s i u i − u j · ∂ h ∂ u j + 2 s j u j − u i · ∂ h ∂ u i + ∂ g α ∂ u i ∂ g β ∂ u j − ∂ g α ∂ u j ∂ g β ∂ u i u i − u j , i, j = 1 , ..., n, i 6 = j, (43) and ∂ 2 h ∂ u i ∂ u i = −  1 + 2 n +2 X j =1 s j  2 s i u i ( u i − 1) h + 2 s i u i ( u i − 1) n X j 6 = i u j ( u j − 1) u j − u i · ∂ h ∂ u j +  n X j 6 = i 2 s j u i − u j + 2 s i + 2 s n +1 u i + 2 s i + 2 s n +2 u i − 1  ∂ h ∂ u i − (44) X j 6 = i ∂ g α ∂ u i ∂ g β ∂ u j − ∂ g α ∂ u j ∂ g β ∂ u i u i − u j · u j ( u j − 1) u i ( u i − 1) − (1 + s 1 + ... + s n +2 ) ∂ g α ∂ u i g β − ∂ g β ∂ u i g α u i ( u i − 1) . Define new c o or dinates v 1 , ..., v n and the functions F α,β ( v 1 , ..., v n ) by v i = h i ( u 1 , ..., u n ) h 0 ( u 1 , ..., u n ) , F α,β = f α,β h 0 ( u 1 , ..., u n ) . (45) Then, in c o or dinates v 1 , ..., v n , the system (42) takes the Go dunov form  ∂ F r,s ∂ v i  t q +  ∂ F s,q ∂ v i  t r +  ∂ F q ,r ∂ v i  t s = 0 , i = 1 , . . . , n. (46) Pro of: Substituting (45) in to (46) and calculating deriv atives of f α,β b y virtue of (43), (44) w e obtain conserv ation la ws of the system (42) as found in [29]. More precisely , the left hand side of (46) is equal to n X j =1 ( − 1) i + j det W i,j u j ( u j − 1) det W R j 27 where R j is the left hand side of (42), the matrix W is defined as W = ( w α,β ) where w 0 ,β = h β ( u 1 , ..., u n ) , w α,β = ∂ h β ∂ u α if α 6 = 0 and W i,j is the n × n -minor of W obtained b y eliminati ng a column with h i and a ro w with ∂ ∂ u j . In other w ords, the left hand side of (46) is equal to n X j =1 ( W − 1 ) i,j u j ( u j − 1) R j , so that (46) holds iden tically mo dulo (42). Remark 1. The sys tem (46) p ossesses a natural action of the group GL n +1 × GL n +1 . Namely , the first cop y of GL n +1 acts on the times t 0 , ..., t n , and in the s ame w a y on the basis g 0 , ..., g n . This action corresp onds to a c hange of basis g 0 , ..., g n in H s 1 ,...,s n +2 . The second copy of GL n +1 acts according to the Prop osition 1. This action corresp onds to a c hange of basis h 0 , ..., h n in H 2 s 1 ,..., 2 s n +2 . Remark 2. Note that F α,β is defined up to an arbitrary linear com bination of 1 , v 1 , ..., v n and, therefore, f α,β is defined up to an arbitrary linear com bination of h 0 , h 1 , ..., h n . This explains wh y f α,β satisfies a linear non-homogeneous system with the same homogeneous part as for elemen ts from H 2 s 1 ,..., 2 s n +2 . Moreo ve r, the structure of the non-homo geneous part of the system (43), (44) con taining linear com binations of ∂ g α ∂ u i ∂ g β ∂ u j − ∂ g α ∂ u j ∂ g β ∂ u i and ∂ g α ∂ u i g β − ∂ g β ∂ u i g α is dictated by the action of GL n +1 on the times t 0 , ..., t n , and in the same w a y on g 0 , ..., g n , compare with (42). Remark 3. It wa s prov en in [27] that an y 2-comp onen t integra ble sy stem is isomorphic to a mem b er of the family (42) with n = 2 , or its appropriat e limit. Therefore, Theorem 5 gives, in particular, a description of the G o duno v form for any generic 2-comp onen ts in tegrable sy stem. If n > 2 there exist n -comp onen t in tegrable s ystems whic h do not b elong to the family (42) or its degeneration s (see, for ex ample, [30]). Ho we v er, n -comp onen t sy stems constructed in [30]) ha ve n conserv ation la ws only and, therefore , do not p ossess a Go duno v form. Probably , the only in tegrable s y stems p ossess ing Go duno v ’s form should b elong to the family (42) or its appropriate degeneration s. 5.3 Application to integrable p oten t ials H(V, W) Let us no w apply these results to the system (11 ). Set n = 2 , choose a triple of indices s = 0 , q = 1 , r = 2 and set t 0 = − t, t 1 = x, t 2 = y where t, x, y are the indep enden t v ariables in (11). Note that the bases in H s 1 ,...,s n +2 and H 2 s 1 ,..., 2 s n +2 are no longer independen t since we ha v e constrain ts on the functions F 1 , 2 , F 2 , 0 , F 0 , 1 , namely , F 2 , 0 = v 1 F 1 , 2 , F 0 , 1 = v 2 F 1 , 2 . T o obtain (11) it remains to set F 1 , 2 = H , v 1 = V , v 2 = W . Therefore, f 1 , 2 = H h 0 , f 2 , 0 = H h 1 , f 0 , 1 = H h 2 . Moreo ver, w e m ust ha ve h i = a ( u 1 , u 2 )( g j g k ,u 1 − g k g j,u 1 ) + b ( u 1 , u 2 )( g j g k ,u 2 − g k g j,u 2 ) + c ( u 1 , u 2 )( g j,u 2 g k ,u 1 − g j,u 1 g k ,u 2 ) where i, j, k is a cyclic p erm utation of 0 , 1 , 2 . Indeed, h i m ust hav e the same structure in g 0 , g 1 , g 2 as co efficien ts at u j,t i in (42 ). Substituting the expressions for h 0 , h 1 , h 2 and f 1 , 2 , f 2 , 0 , f 0 , 1 in to the equations for H 2 s 1 ,..., 2 s n +2 and (43), (44), resp ectiv ely , and using the equations (40), (41) for g 0 , g 1 , g 2 , we obtain that s 1 = s 2 = − s 3 = s 4 = − 1 3 , along with the follo wing expressions for the functions h 0 , h 1 , h 2 : h 0 ( u 1 , u 2 ) = C  u 2 u 2 − 1 ( g 1 g 2 ,u 1 − g 2 g 1 ,u 1 )+ 28 u 1 u 1 − 1 ( g 1 g 2 ,u 2 − g 2 g 1 ,u 2 ) + 3( u 1 − u 2 )( g 1 ,u 2 g 2 ,u 1 − g 1 ,u 1 g 2 ,u 2 )  , h 1 ( u 1 , u 2 ) = C  u 2 u 2 − 1 ( g 2 g 0 ,u 1 − g 0 g 2 ,u 1 )+ u 1 u 1 − 1 ( g 2 g 0 ,u 2 − g 0 g 2 ,u 2 ) + 3( u 1 − u 2 )( g 2 ,u 2 g 0 ,u 1 − g 2 ,u 1 g 0 ,u 2 )  , h 2 ( u 1 , u 2 ) = C  u 2 u 2 − 1 ( g 0 g 1 ,u 1 − g 1 g 0 ,u 1 )+ u 1 u 1 − 1 ( g 0 g 1 ,u 2 − g 1 g 0 ,u 2 ) + 3( u 1 − u 2 )( g 0 ,u 2 g 1 ,u 1 − g 0 ,u 1 g 1 ,u 2 )  , where C = ( u 1 − 1) 2 / 3 ( u 2 − 1) 2 / 3 ( u 1 − u 2 ) − 1 / 3 . W e also obtain the follo wing system for H , H u 1 = u 2 − 1 C ( u 1 − u 2 ) , H u 2 = u 1 − 1 C ( u 2 − u 1 ) . The solution of this system reads H ( u 1 , u 2 ) = G  u 1 − 1 u 2 − 1  where G ′ ( t ) = 1 t 2 / 3 ( t − 1) 2 / 3 . Summarizing, w e obtain the pro of of the follo wing Theorem 4 L et the p otential H ( V , W ) b e define d p ar ametric al ly in the form H = G  u 1 − 1 u 2 − 1  , V = h 1 h 0 , W = h 2 h 0 , wher e G ′ ( t ) = 1 t 2 / 3 ( t − 1) 2 / 3 and h 0 , h 1 , h 2 ar e thr e e line arly indep endent solution of the hyp er ge o- metric system u 1 (1 − u 1 ) h u 1 ,u 1 − 4 3 u 1 h u 1 − 2 9 h = 2 3 u 1 ( u 1 − 1) u 1 − u 2 h u 1 + 2 3 u 2 ( u 2 − 1) u 2 − u 1 h u 2 , u 2 (1 − u 2 ) h u 2 ,u 2 − 4 3 u 2 h u 2 − 2 9 h = 2 3 u 1 ( u 1 − 1) u 1 − u 2 h u 1 + 2 3 u 2 ( u 2 − 1) u 2 − u 1 h u 2 , h u 1 ,u 2 = 2 3 h u 2 − h u 1 u 2 − u 1 ; (this system c oincides with (40), (41) for the values of c onstants s 1 = s 2 = − s 3 = s 4 = − 2 3 ). Then H ( V , W ) pr ovides the generic solution to the system (27). In p articular, it do es not p ossess any c ontinuous symmetry fr om the e quivalenc e gr oup. 29 6 App endix. The m etho d of h ydro dynamic reductions. Deriv a- tion of the in teg rabilit y conditions Applied to a (2 + 1) -dimensional s ystem (1), the meth o d of h ydro dynamic reductions consists of seeking multi -phase solutions in the form u ( x, y , t ) = u ( R 1 , ..., R n ) where the phases R i ( x, y , t ) are required to s atisf y a pair of (1 + 1) -dimensional systems of h ydro dynamic t y pe, R i t = λ i ( R ) R i x , R i y = µ i ( R ) R i x . Solutions of this form, kno wn as ‘non-linear in teractions of n planar s imple wa v es’ [36, 7, 35], ha ve b een extensiv ely discussed in gas dynamics; later, they reappeared in the con text of the disp ersionless KP hierarc h y , see [19, 20] and referenc es therein. Effectiv ely , one decouples the (2+ 1) -dimensional system (1) into a pair of comm uting n -componen t (1 +1) -dimensional systems, whic h are called its hydrodynamic reductions. Substitut ing the ansatz u ( R 1 , ..., R n ) in to (1) one obtains ( λ i I n − A − µ i B ) ∂ i u = 0 , i = 1 , ..., n, (47) ∂ i = ∂ /∂ R i , implying that b oth c haracteristic sp eeds λ i and µ i satisfy the disp ersion relation det( λI n − A − µB ) = 0 , (48) whic h defines an algebraic curv e of degree n on the ( λ, µ ) -plane. Moreo v er, λ i and µ i are required to satisf y the comm utativit y conditions ∂ j λ i λ j − λ i = ∂ j µ i µ j − µ i , (49) i 6 = j , see [38]. I n w as observ ed in [13] that the requiremen t of the existence of ‘s ufficien tly man y’ h ydro dynamic reductions imp oses strong restrictions on the system (1), and provide s an efficien t classification criterion. T o b e precise, w e will call a system (1) in tegrable if, for an y n , it p ossesses infinitely man y n -comp onen t hydrodynamic reductions parametrised b y n arbitrary functions of a single v ariable. Th us, in tegrable systems are required to poss ess an infinit y of n -phase solutions whic h can b e view ed as natural disp ersionless analogues of algebro-geomet ric solutions of soliton equations. In this A ppendix w e illustrate the ab o ve pro cedure by c haracterising all Hamiltonians h ( v , w ) for whic h the system (7) is in tegrable b y the metho d of hydrodynamic reductions. Rewriting Eqs. (7) as v t = h vw v x + h w w w x , w t = h vv v x + h vw w x + h vw v y + h w w w y , (50) w e seek n-phase solutions in the form v = v ( R 1 , R 2 , . . . , R n ) , w = w ( R 1 , R 2 , . . . , R n ) where the phases (R iemann in v arian ts) R i satisfy the equations R i t = λ i ( R ) R i x , R i y = µ i ( R ) R i x . The substitution of this ansatz into (50) implies the relations v i λ i = h vw v i + h w w w i , w i λ i = h vv v i + h vw w i + h vw v i µ i + h w w w i µ i , (51) 30 here v i = ∂ i v , w i = ∂ i w, ∂ i = ∂ /∂ R i . The condition of their non-trivial s olv abilit y implies the disp ersion relation for λ i and µ i , − λ i 2 + h w w λ i µ i + 2 h vw λ i + h vv h w w − h 2 vw = 0 . In what follow s we assume that the disp ersion relation defines an irreducible conic, which is equiv alen t to the requiremen t h w w 6 = 0 , h vv h w w − h 2 vw 6 = 0 . Setting v i = φ i w i w e can rewrite (51) in the from φ i λ i = h vw φ i + h w w , λ i = h vv φ i + h vw µ i φ i + h w w µ i + h vw . (52) W e also require the compatib ilit y of the relations v i = φ i w i , which give s ∂ i ∂ j w = ∂ j φ i φ j − φ i ∂ i w + ∂ i φ j φ i − φ j ∂ j w. (53) Expressing λ i , µ i in terms of φ i from (52 ), λ i = h vw φ i + h w w φ i , µ i = h w w − h vv φ i 2 φ i ( h vw φ i + h w w ) , and substituting these expressions in to the comm utativit y conditions (49) w e obtain relations of the form ∂ j φ i = ( . . . ) ∂ j w , ∂ i ∂ j w = ( . . . ) ∂ i w∂ j w where dots denote certain rational expressions in φ i , φ j whose co efficien ts dep end on the Hamiltonian densit y h ( v , w ) and its deriv atives up to the order three. The compatibil it y conditions ∂ j ∂ k φ i = ∂ k ∂ j φ i tak e the form P ∂ j w∂ k w = 0 where P is a p olynomial in φ i , φ j , φ k . Setting all co efficien ts of this p olynomial equal to zero one obtains the in tegrabilit y conditions (16). A c kno wl edgemen ts W e thank K Kh usn utdino v a, O Mokho v and M P a vlo v for clarifying discussions. The researc h of EVF was partially supp orted b y the European R esearc h Council A dv anced Gran t F roM-PDE. References [1] M.J. Ablo witz and P .A. 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