On non-Abelian Toda $A_2^{(1)}$ model and related hierarchies

On non-Ab elian T o da A (1) 2 mo del and related hierarc hies Dmitry K. Demsk oi, Jyh-Hao Lee Institute of Mathematic s, Academia S inica, T aip ei, T aiw an Abstract W e study li miting cases of the t w o kno wn in tegrable c hiral-t yp e mo dels with tree-dimensional configuration space. On e of the in itial m o dels is the non-Ab elian T o da A (1) 2 mo del and the other w as found by means of the symmetry approac h by A.G. Meshk o v and one of the authors. The C- in tegrabilit y of the reduced mo dels is established by constructing their complete sets of integral s and general so lutions. A description of the generalized symmetry algebras of these mo d els is giv en in terms of op erators mapp ing int egrals in to symmetries. The integ rals of the Liouville- t yp e systems are kno wn to define Miura-t yp e transformations for their generalized symmetries. This fact allo w ed u s to fin d a few new systems of the Y a jima-Oik aw a t yp e. W e present a recursion op erator for one th em. I. Intro duct i o n It is known that many S-integrable h yp erb olic equations ha v e limiting cases whic h a r e in tegrable explicitly (see e.g. [1]). The latter hav e a few other c haracteristic prop erties suc h as pr esence of generalized sym metries and non trivial in tegra ls (ps eudo constants ). These e quations constitute a sub class of C-in tegrable equations and called the Liouville-t yp e equations. It app ears that t he ab o v e men tioned in terrelation b etw een S- a nd C-in tegrable equations [2] can be used for establishing links b et w een differen t thoug h related hierarc hies and also for studying their pro p erties [3, 4]. Belo w w e consider in tegrable models with Lagrangians of the form L = u x u t + η v x w t + f , (1) where subscripts denote part ia l deriv ativ es, η and f are functions of the field v ariables u , v, and w . Suc h mo dels are sometimes referred as chiral-t yp e mo dels [5, 6]. The general form of the c hiral-ty p e Lagrangian is L = g ij ( u ) u i x u j t + f ( u ) , (2) where g is some non- degenerate matrix (metric tensor). Throughout the article w e assume summation o v er the rep eated indices. The no n-Ab elian T o da A (1) 2 mo del has the Lagrangian L 1 = u t u x + 4 3 v x w t v w + e u + a  v w + 3 4 e u  e u + be − 2 u , (3) where a and b are arbitra ry constan ts. A long with (3) w e a lso consider a mo del with L 2 = u t u x + 4 v x w t v w + c + av e u + bw e − u . (4) The reas on wh y these models are considered in one pap er is that they b oth related to the Y a jima- Oik aw a hierarc h y and also b ecause of the ob vious similarit y b et w een them. The other common 1 feature is that they b oth admit generalized sym metries whic h are p olynom ial w.r.t. deriv ativ es of field v ariables. The in tegrable mo del corresp onding to (3) w as derive d by means of a general algebraic construction of affine non-Ab elian T o da mo dels [7]. The simple st non-Ab elian T o da A (1) 1 mo del is actually the w ell kno wn Lund-Regg e system (complex sine-Gordo n I equation) [8]. The mo del corr esp o nding to L 2 w as studied in [9], where it was sho wn to ha v e a La x represen ta t io n and infin itely man y conserv ed densities. Obvious ly this mo del can b e view ed as one of the extens ions of the classical sine-Gor do n mo del. The limiting cases of (4) with a = 0 or b = 0 we re considered in [3]. It was prov ed that in either of these t w o instances the mo del has a complete set of in tegrals a nd hence can b e integrated explicitly . The actual general solution will b e giv en in Section IV of this pap er. W e also show that t hese reduced models are related to the op en T o da A 2 c hain. One of the ob jectiv es o f this article is to establish connections b et w een hierarc hies o f generalized symmetries [10] generated b y (3) and (4). W e show that they are b oth related to the Y a jima-Oik aw a hierarc h y [11] by means of Miura-type transformations generated by the integrals of the reduced systems . The pap er is organized as follows. First, w e consider the reduced models deriv ed from (3) a nd (4), construct their complete sets of in tegrals, proving therefore that they are of t he Liouville t yp e. The generalized Laplace in v ariants are used here as an auxiliary to ol assisting in construction of the in tegrals. Section I I I is dev oted to finding the op erators mapping in tegrals of systems in question in to their generalized symmetries. In Section IV w e sho w that solutions of the reduced systems are related to solutions of either the Lio uville equation or the op en T o da A 2 c hain. This enables us to giv e explicit form ulas of general solutions that are free o f quadratures. In the last section in tegrals of the reduced systems are used to find integrable ev olution systems related to the simplest generalized symmetries o f (3) and (4). I I. In tegrals and generalized Laplace in v arian ts of reduced systems The Laplace in v ariants of systems of equations ha v e previously b een considered in a few differen t con texts [12, 1, 1 3, 14]. It has b een established [15] that the chains of the Laplace inv ariants for the most w ell-kno wn h yp erb olic systems having complete sets of in tegrals – the op en T o da c hains, are finite. As w e announced b efore the reduced systems considered in this articles are related to the op en T o da A 2 c hain. Therefore the Laplace inv a rian ts for these systems mus t hav e similar prop erties. There is, ho w ev er, a difference: w e will sho w that for systems with Lagrangians (3), (4) under condition a = 0 ( b = 0), the c hains of the Lapla ce inv a rian ts H k terminate in classical sense, i.e. det H k 6 = 0 , H k +1 = 0 for some k . The Laplace in v a rian ts H k for the op en T o da ch ains are known to b ecome degenerate though not iden tical zeros for some k (see e.g. [1]). The L a place inv aria n ts for the syste m of hyperb olic equations u i tx = F i ( u, u t , u x ) , u = ( u 1 , . . . , u n ) (5) are intro duced as follo ws. First, w e consider the linearized system S i tx − ∂ F i ∂ u j x S j x − ∂ F i ∂ u j t S j t − ∂ F i ∂ u j S j = 0 . (6) The fir st Laplace in v arian ts H − 1 , H 0 of (6) are defined as (see [1] fo r detailed exposition) ( H 0 ) i j = ∂ F i ∂ u k t ∂ F k ∂ u j x + ∂ F i ∂ u j − D x  ∂ F i ∂ u j x  , ( H − 1 ) i j = ∂ F i ∂ u k x ∂ F k ∂ u j t + ∂ F i ∂ u j − D t  ∂ F i ∂ u j t  , 2 where the total deriv ativ e op erator s D x , D t w.r.t correspo nding v ariables are calculated in virtue of system (5). The ch ain of t he Laplace in v ariants is in tro duced according to the f o llo wing recurren t form ulas A k +1 H k = − D t ( H k ) + H k A k , ( A 0 ) i j = − ∂ F i ∂ u j x , ( H k +1 ) i j = ( H k ) i j + D x ( A k +1 ) i j − ∂ F i ∂ u s t ( A k +1 ) s j + ( A k +1 ) i s ∂ F s ∂ u j t + D t ∂ F i ∂ u j t . If det H k 6 = 0, then the matrix A k +1 and hence t he next Laplace inv arian t H k +1 are determined uniquely . If it is true for a ll k , then we hav e an infinite c hain o f the Laplace in v ariants. How ev er, if the system in question admits non trivial in tegrals, then det H k = 0 for some k . Nev ertheless ev en in this case the chain can be con tin ued if certain conditions are satisfied [1]. The first Laplace inv arian ts H − 1 , H 0 for systems with Lagrangian (2) under conditio n f = 0 hav e a nice geometric interpre tation, namely they are closely r elat ed to the Riemann curv ature tensor. In fact the Riemann curv a t ur e tensor can b e defined as the first Laplace inv aria nt for the system u i tx + Γ i j k u j x u k t = 0 , (7) where Γ i j k = 1 2 g ( is )  ∂ g j s ∂ u k + ∂ g sk ∂ u j − ∂ g j k ∂ u s  , g ( is ) = 1 2 ( g is + g si ) Note that Γ i j k is not necessarily symme tric. It is no t difficult to find that fo r sys tem (7) the Laplace in v ariant H 0 is given explicitly b y ( H 0 ) i j =  ∂ Γ i j s ∂ u k − ∂ Γ i k s ∂ u j + Γ i k p Γ p j s − Γ i j p Γ p k s  u k x u s t = R i sk j u k x u s t , where R i sk j is the curv ature tensor corresp onding to g ij . It immediately follows from this that det H 0 = 0. Indeed, due to the an tisymm etry of R i sk j w.r.t. indices k , j w e ha v e R i sk j u s t u k x u j x = 0. This is a reflection of the w ell kno wn fact that an y system (7) has the first order in tegral ω = g ij u i x u j x , D t ω = 0 . (8) Generally we conjecture tha t if n > k 0 , then a coupled syste m of fo rm (5) admits n − k 0 first order in tegrals. F urthermore if rank H i = k i and k i − 1 > k i for i > 0, then system (5) admits k i − 1 − k i in tegrals of the order i + 1. This statemen t has b een verifie d for differen t Liouville-ty p e systems and in particular for the op en T o da c hains [15]. W e ha v e a lso v erified the v a lidity of this statemen t for the reduced systems in question. The explicit form o f the system corresp o nding to Lagrangian (3) is u tx = − 2 3 ψ 2 v x w t e u + 1 2 a e u ( 3 2 e u + v w ) − b e − 2 u , v tx = ψ v x ( w v t + e u u t ) + 3 4 aψ − 1 v e u , w tx = ψw t ( w x v + e u u x ) + 3 4 aψ − 1 w e u , (9) where ψ = ( v w + e u ) − 1 . Belo w we consider the r educed systems derive d from (9) b y successiv ely setting a = b = 0, a = 0, and b = 0. The corresp onding hyperb o lic systems will b e referred as S 1 , S b 1 , and S a 1 . Systems S a 1 and S b 1 are also kno wn as the r educed A (2) 2 Bershadsky-P olyak ov and A (1 , 1) 2 non-Ab elian T o da mo dels (see [16] and references therein). Our ob jectiv e is to sho w that these systems hav e terminating sequenc es o f the Lapla ce in v ariants and complete sets of in tegrals. The system corresp onding to (4) has the form u tx = a 2 v e u − b 2 w e − u , v tx = b 4 ϕ − 1 e − u + ϕw v x v t , w tx = a 4 ϕ − 1 e − u + ϕv w t w x , (10) 3 where ϕ = ( v w + c ) − 1 . The reduced systems deriv ed from (10) w ere considered in [3] where they w ere sho wn to b elong to a class of Liouville-t yp e systems. By analog y with t he previous systems they will b e referred as S 2 , S b 2 , and S a 2 . Note that systems (9) and (10) admit the follo wing symmetries t → x, x → t, v → w , w → v and t → x, x → t corresp ondingly . This allo ws one t o construct t − in tegrals from x − in tegrals and vice v ersa if either of them is kno wn. W e start with the fully reduce d system S 1 ( a = b = 0) u tx = − 2 3 ψ 2 e u v x w t , v tx = ψ v x ( e u u t + v t w ) , w tx = ψw t ( e u u x + w x v ) . (11) It is not difficult to c hec k that for this sy stem we hav e rank H 0 = 1 and H 1 H 0 = 0, therefore w e conjecture that (1 1 ) has tw o first order and one second order integrals. T o derive the complete set of in tegrals for this system, w e used the pro cedure suggested in [3 ]. F or the applicabilit y of the pro cedure one needs to hav e a nontrivial in tegral and a non- degenerate higher comm uting flow (symmetry). The no ntrivial integral for ( 1 1) has the form (8). The simplest generalized symmetry is common for all systems derived from (3) and giv en by formula (25 ) (see b elo w). It is con v enien t to write in tegrals in terms of the follo wing quantities α = u, β = v x w ψ , γ = ln w . (12) Then t he in tegrals of (11) can b e written as m = α x + 2 3 β , p = 4 3 β ( α x − γ x + 1 3 β ) , q = 2( α x + γ x − 1 3 β − β − 1 β x ) . (13) Note tha t inte gral (8) can b e brough t into the for m ω = m 2 − p . System S b 1 ( a = 0): u tx = − 2 3 ψ 2 e u v x w t − be − 2 u , v tx = ψv x ( e u u t + v t w ) , w tx = ψw t ( e u u x + w x v ) . (14) F or this system w e ha v e rank H 0 = 3 and H 1 = 0 so w e lo ok for the three indep endent sec ond order in tegrals. Because system S 1 app ears to b e a limiting case o f S a 1 and S b 1 the in tegrals of the la tter can b e expresse d in terms o f integrals of the former. By direct calculation it is not difficult to find that the in tegrals are µ = m − 1 2 q , ν = m x + m 2 − p, λ = − 2 p x − q p. (15) System S a 1 ( b = 0): u tx = − 2 3 ψ 2 e u v x w t + a 4 e u ( e u + 2 ψ − 1 ) , v tx = ψ v x ( e u u t + w v t ) + 3 4 av e u ψ − 1 , w tx = ψw t ( e u u x + w x v ) + 3 4 aw e u ψ − 1 . (16) As in the previous case we hav e rank H 0 = 3, H 1 = 0 and thus the sys tem has three second order in tegrals g iv en explic itly by ρ = m 2 − m x − p, θ = p q , φ = p x p − 1 + 1 2 q − m. (17) The complete set of in tegra ls for S a 2 and hence for S b 2 w ere constructed in [3]. As in the previous case it is con v enien t to introduce the quan tities α = u, β = v x w ϕ, γ = ln w . (18) System S 2 decouples into the d’Alam bert equation and the reduced Lund- Regge system u tx = 0 , v tx = w ϕv t v x , w tx = v ϕw t w x . (19) 4 The integrals are m = − 1 2 α x , p = − β γ x , q = 2( γ x − α x − β x β − 1 − β ) . (20) System (19 ) w as used in [17, 18] as the w orking example of a Liouville-t yp e system. System S a 2 is given by u tx = a 2 v e u , v tx = ϕw v x v t , w tx = a 4 ψ − 1 e u + ϕv w t w x . (21) The complete set of integrals for this system has the f orm (15) w ith m, p, and q giv en b y (20). System S b 2 is obtained from (2 1) b y means of the transformation u → − u, v → w , w → v , a → b. The in tegrals for this system ha v e the form (17) with m, p, and q g iv en b y (20). W e w ould like to p oin t out that the presen ted sets o f inte grals are minimal . I I I. The s t ructure of generalize d symmetries Higher symmetries of the Liouville-t yp e systems are kno wn to hav e the sp ecial structure S = M ω , where M is some linear differen tial op erator and ω a v ector-function of in tegrals. F unction S is assumed to satisfy equation (6) . Op erator M giv es a complete description of symmetry algebra for a give n Liouville-t yp e h yp erb olic system. It satisfies the follow ing o p erator equation ( D x D t − F ∗ ) M = T D t , (22) where T is some differen tial op erator and F ∗ stands for the F reshet deriv ativ e of the right hand side of (5). In principle, relatio n (22) can b e used to find op erator M , but it app ears more con v enien t to use results of [18 ] where it w as prov ed that fo r any Liouville-t yp e syste m of the fo rm (5) there exists a differen tial op erato r P suc h t ha t ω + ∗ = ( − D x + F u t ) + ◦ P , (23) where ω + ∗ stands for the op erator fo rmally conj ug ated to ω ∗ . Then a ccording t o [19] the op erator M = g − 1 s P maps integrals of (5) in to its symmetries , where g s is the symmetric part of the metric tensor. The matrix g − 1 s for mo dels S 1 , S a 1 , a nd S b 1 has the form g − 1 s =   1 0 0 0 0 3 2 ψ − 1 0 3 2 ψ − 1 0   . Ha ving found in tegra ls fo r thes e mo dels, it is not difficult to factorize opera t o r ω + ∗ as in fo rm ula (23), and th us to find the op erator P . Therefore w e ha v e found the following M − op erator f or mo del (11) M = g − 1 s P =   1 − 2 3 ψ w v x 2 0 v x 3 v w w x − w u x − 2 3 w 2 v x ψ 3( ψ v x ) − 1 D x − w   . W e denote M a and M b the M-op erators for mo dels S a 1 and S b 1 corresp ondingly . These op erators can b e factorized in the following w a y M a = MF a , M b = MF b (24) 5 where F b =   − 2 D x − 2 m 0 0 − 2 D x − 1 2 q 1 0 1 4 p   , F a =   D x + 2 m 0 1 2 2 q − 2 p − 1 D x 0 − p − 1 2   . If w e denote a s A 1 , A a 1 , a nd A b 1 the generalized symmetries alg ebras of systems S 1 , S a 1 , and S b 1 corresp ondingly , then from (24) the f o llo wing relation fo llows A a 1 ⊂ A 1 , A b 1 ⊂ A 1 . Finally w e would lik e to p oint out the the simplest generalized symme try of (9) can b e written in the form   u t v t w t   = M a   0 1 0   = M b   0 0 − 4   . (25) It is easy to c hec k that (25) is the second order p olynomial (w.r.t deriv atives ) evolutionary system. IV. General solutions of the redu c e d systems In [20] a reduction pro cedure was used t o find a general solution of t he open T o da A n c hain. The idea is to replace a system in question b y an equiv alen t hig her order PDE whic h can then b e in tegrated explicitly . Here a similar pro cedure is applied to systems S a i , S b i , a nd S i ( i = 1 , 2 ). The solutions of these systems will b e given in quadrat ure- free form. W e start with the simplest system S 1 whic h is g iv en b y ( 11). First, expressing u t and u x from the second and third equations corresp ondingly and t hen calculating the compatibility condition u tx = u xt w e find h tx h − h t h x = 0 , where h = w t v − 1 x . This a llo ws us to pa r a metrize functions v and w the follo wing w ay v = T 1 e s s t , w = − X 1 e s s x . Here and b elo w T i ( t ) and X i ( x ) are arbitrary functions of the indicated v ariables. Now conside ring equations (11) 2 , (11) 3 as OD Es (w.r.t. to u ) w e find u = ln( T 1 X 1 ) + 2 s + ln ( − s tx ) . Substituting this expression in to (11) 1 w e find that s satisfies the equation s txx s ttx − s ttxx s tx = 8 3 s 3 tx . (26) Again the substitution s tx = − exp( r ) reduces (26) to the L io uville equation r tx = 8 3 e r ha ving t he w ell-kno wn general solutio n r = ln  3 4 X ′ 2 T ′ 2 ( X 2 + T 2 ) 2  . Th us we hav e finally s = 3 4 ln( X 2 + T 2 ) + T 3 + X 3 . Therefore the general solution of (11) is u = 2( T 3 + X 3 ) + ln  3 4 T 1 X 1 X ′ 2 T ′ 2 √ X 2 + T 2  , v = T 1 exp( X 3 + T 3 )( X 2 + T 2 ) 3 4  3 4 T ′ 2 X 2 + T 2 + T ′ 3  , w = − X 1 exp( X 3 + T 3 )( X 2 + T 2 ) 3 4  3 4 X ′ 2 X 2 + T 2 + X ′ 3  . 6 The same reduction pro cedure can b e applied to system S b 1 . The differenc e is that instead of (26) one gets the fo llowing system s txx s ttx − s ttxx s tx − 8 3 s 3 tx = b exp( r − 4 s ) , r tx = 0 whic h in turn is equiv alen t to s tx = exp( − 8 3 s + τ ) , τ tx = − b exp( 4 3 s − 2 τ + r ) , r tx = 0 . It is easy to see tha t the latter system is reducible to the op en A 2 T o da c hain by the transfor mat ion s → 3 / 4 s − 3 / 20 r , τ → τ − 2 / 5 r . Using this connection [20] one can express the general solution of S b 1 in the form u = ln( r t r x ) + log( − s tx ) + 2 s, v = r t exp( s ) s t , w = − r x exp( s ) s x , where r = ln  s txx s ttx − s ttxx s tx − 8 3 s 3 tx  + 4 s − log ( b ) , s = 3 4 ln( X 1 T 1 + X 2 T 2 + X 3 T 3 ) . (27) The general solution of the mo del S a 1 (with a = − 4 / 3 ) can b e obt a ined from the solutio n of S b 1 b y using the transformation [16] u → − u − 1 2 ln  1 + 4 3 e − u v w  , v → w e − u (1 + 4 3 e − u v w ) − 1 / 4 , w = 4 3 v e − u (1 + 4 3 e − u v w ) − 1 / 4 . No w consider the reduced systems S 2 and S a 2 , i.e. (19) and (21). The general solution of S 2 can b e found the fo llo wing wa y . First, w e express w from the first equation in (19) w = v tx ( v tx v − v t v x ) − 1 and t hen substituting it in to the second equation, w e get det   v v t v tt v x v tx v ttx v xx v txx v ttxx   = 0 . (28) The la tter equation has the follo wing g eneral solution v = X 1 T 1 + X 2 T 2 , a nd thus w e ha v e w = ( X ′ 1 T ′ 1 + X ′ 2 T ′ 2 ) ( T 2 T ′ 1 − T 1 T ′ 2 )( X 1 X ′ 2 − X 2 X ′ 1 ) . No w w e turn to system S a 2 . The v ariables v and w can b e expressed from the first and second equations in (21), this giv es v = 2 a − 1 exp  − s 2  , w = ac 4 exp  s 2  ( s x s t − 2 s tx ) s − 1 tx , (29) where s = 2 u − 2 ln u tx . Substituting (29) to (21) 3 yields s ttxx s tx − s ttx s txx − s 3 tx = − 1 2 s 2 tx exp  u − 1 2 s  . (30) If w e in tro duce the new quan tit y r = − 2 ln( s tx ) − u + 2 s , then one can c hec k tha t r tx = 0. Therefore system (21) is equiv alent to the Op en T o da A 2 c hain coupled with the d’Alambert equation u tx = ex p  u − 1 2 s  , s tx = exp( − 1 2 u + s − 1 2 r ) , r tx = 0 . (31) This enables us to express the general solutio n of (31) in the fo rm u = − 2 log Q, v = 4 a ( Q x Q t − Q tx Q ) , w = − ac 4 Q ttxx Q − Q tt Q xx Q ( Q xx Q tt Q tx + Q x Q t Q ttxx − QQ tx Q ttxx − Q txx Q x Q tt − Q ttx Q t Q xx + Q ttx QQ txx ) , (32) where Q = X 1 T 1 + X 2 T 2 + X 3 T 3 . 7 V. D ifferen tial substit utions and mo difie d Y a jim a-Oik a w a s ys- tems It is well kno wn [1 ] that the minimal integrals of Lio uville-t yp e systems define differen tial substitu- tions for their generalized symmetries. Therefore having constructed t hem for S i , S a i , and S b i w e also found the differen tial substitutions for generalized symmetries of these systems. No w it is easy task to construct corresp onding mo dified ev olutionary systems fo r (25), but first let us rewrite (25) in v ariables (12): α τ = − 2 3 β x + 4 3 β α x , β τ = β xx − 2( γ x β ) x + 4 3 β x β , γ τ = − γ xx + α xx + α 2 x − γ 2 x − 1 3 β 2 + 4 3 β γ x . (33) The second system in the chain of t r ansformed systems corresp onds to (1 3) and is giv en b y m τ = p x , p τ = − p xx − ( pq ) x + 2 mp x , q τ = q xx + 2( mq ) x − 1 2 ( q 2 ) x . (34) The integrals of S a 1 and S b 1 tak e syste m (34) in to ρ τ = θ x , θ τ = θ xx − 2( φθ ) x , φ τ = − φ xx − ( φ 2 − ρ ) x , (35) ν τ = − λ x , λ τ = − λ xx + 2( λµ ) x , µ τ = µ xx + ( µ 2 − ν ) x (36) corresp ondingly . Note that systems (35) and (36) a re related b y the transformation τ → − τ . System (35) is kno wn (see e.g. [3]) to b e related to the Y a jima-Oik aw a system [11] U τ = ( V W ) x , V τ = − V xx + U V , W τ = W xx − U W (37) b y the transformation ρ = U, θ = V W , φ = V x /V . (38) On t he other hand, it is kno wn [3] that the simplest generalized symmetry u τ = 2 ϕv x w x , v τ = v xx − 2 ϕv v x w x + u x v x , w τ = − w xx + 2 ϕw v x w x + u x w x (39) of (10) is related to (34) b y means of the in tegrals of S 2 . Note that system (39) in v ariables α , β , and γ has the simple p olynomial form [5] α τ = 2 β γ x , β τ = β xx + ( β 2 + β α x − 2 β γ x ) x , γ τ = − γ xx + α x γ x − γ 2 x + 2 β γ x . (40) The evolution systems listed ab o v e constitute a sub class o f mo dified Y a jima- Oik aw a systems sin- gled out by the relation to h yp erb olic systems with Lag r a ngians (3 ) and (4). Their integrabilit y ob viously follo ws fr o m the inte grability of the Y a jima-Oik a w a system. Man y of their prop erties lik e bi-Hamiltonian structure, recursion op erato r s, etc, can b e obtained from the ones of the Y a jima- Oik aw a system itself. The most in teresting of them is pro bably system (34) as it is related to b oth 8 hierarc hies of systems with (3) and (4). W e found its recursion op erator in the form R =   1 4 0 0 0 1 0 0 0 1   D 2 x +   0 − 3 4 0 0 q − 2 m 0 3 2 q 0 2 m − q   D x +   p − m 2 1 2 m − 1 4 q − 3 4 p − 3 2 pq − 3 2 p x p − mq + 1 2 q x + 1 4 q 2 p x − 2 mp + p q − 1 2 q 2 + mq + 5 2 q x 3 2 q − 3 2 q x + 2 m x + p − mq + 1 4 q 2   −   m x p x q x   D − 1 x  m − 1 2 0  +    p x − p xx − p x q − q x p + 2 p x m q xx + 2 q m x + 2 q x m − q q x    D − 1 x  1 0 − 1 2  . (41) One can see that it has standard structure of the nonlo cal part, i.e. it is a pro duct of symmetries and co-symmetries. W e would also lik e to p oin t out that (34 ) is Hamiltonian with the following lo c al Hamiltonian op erator   m t p t q t   = J    δH 1 δm δH 1 δp δH 1 δq    , (42) where J =   − 1 2 D x 0 − 2 D x 0 2 pD x + p x 2 D 2 x + ( q − 4 m ) D x − 2 D x − 2 D 2 x + D x ( q − 4 m ) 0   , (43) H 1 = − 1 2 p q Applying (41) to (43) one can g enerate infinitely many Hamilto nian o p erators. Finally w e w ould lik e to note that in pap er [7] one more system of the Y a jima- O ik aw a t yp e is found. It has the form u t = − ( w v ) x , v t = v xx − v u x − v u 2 − w v 2 , w t = − w xx − w u x + w u 2 + w 2 v . (44) The Miura- t yp e transformation relating (44) with (37) is giv en b y U = u 2 + u x + v w , V = w x + w u, W = − 2 v . Ac kno wledgements Authors ar e grateful to M.V.P a vlo v, O.K.P ashaev, and V.V.Sok olov for fruitful discussions. References [1] Z hib er, A. V., Sok olov , V. V., Exactly in tegrable h yperb olic equations o f Liouville type, Russ. Math. 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