Integrable pseudopotentials related to generalized hypergeometric functions

In tegrable pseudop oten tials related to generalized h yp ergeome tric function s A.V. Odesskii 1 , 2 , V.V. Sok olo v 1 1 L.D. Landau Institute for Theore tical Ph ys ics (Russia) 2 Univ ersity of Manc hes ter (UK) Abstract W e construct in tegrable pseudop oten tials with an arbitra ry n um b er o f fields in terms o f generalized hypergeometric functions. These pseudop oten tials yield some integrable (2+1)-dimensional h ydro dynamic type systems. In t wo particular cases these systems are equiv a len t to integrable scalar 3- dimensional equations of second order. An interes ting class of in tegrable (1+1)-dimensional h ydro dy namic type systems is also g enerated b y our pseudop o ten tials. MSC n umbers: 17B80, 17B63, 32L81, 14H70 Address : L.D. La nda u Institute for Theoretical Ph ysics of Russian Academ y o f Sciences, Kosygina 2, 119334, Mosc o w, Russia E-mail : alexander.o desskii@manc hester.ac.uk, o desskii@itp.ac.ru, sok olo v@itp.ac.ru 1 Con t e n ts 1 In t r o duction 3 2 Generalized h yp er geometric functions 6 3 Pseudop oten t ials of defect 0 8 4 Pseudop oten t ials of defect k > 0 11 5 In t egr able ( 1+1) -dimensiona l systems of h ydro dynamic type 20 6 Hydro dynamic reductions and in t egrabilit y 22 7 Conclusion 25 2 1 In tro duction The main ob j ect of in tegrabilit y theory is the Lax equation L t = [ L, A ] . (1.1) Here A and L are op erators dep ending on functions u 1 , ..., u n and (1.1) is equiv alen t to a system of nonlinear differential equations for u i . F or the KP-hierar ch y and its differen t reductions A is a linear differen tial op erator A = P r i ∂ i x , whose co efficien t s r i are differen tial p o lynomials in u 1 , ..., u n . The L -op erator could b e a differen tial o p erator or a mor e complicated o b ject lik e a ratio of t wo differen t ial op erators or a fo rmal (non- comm utative) La ur ent series with resp ect to ∂ − 1 x . The dispersionless analog of (1.1) has t he following form L t = { L, A } , (1.2) where { L, A } = A p L x − A x L p . As usual, the comm utat o r in (1.1) is replaced by the P oisson brac ke t and the non-comm utativ e v a riable ∂ x b y the commutativ e ”sp ectral” parameter p. The transformation L ( x, t, p ) → p ( x, t, L ) reduces (1.2) to the follow ing conserv a tiv e f o rm p t = A ( p, u 1 , . . . , u n ) x , (1.3) where L plays the role of a parameter. The latter equation can b e rewritten as ψ t = A ( ψ x , u 1 , . . . , u n ) , (1.4) where p = ψ x . Equations ( 1.4) can b e c hosen as a basis, on whic h a theory of integrable 3- dimensional disp ersionless PDEs can b e built. Most suc h equations can b e written in the form n X j =1 a ij ( u ) u j,t 1 + n X j =1 b ij ( u ) u j,t 2 + n X j =1 c ij ( u ) u j,t 3 = 0 , i = 1 , ..., l , (1.5) where u = ( u 1 , . . . , u n ) . All kno wn integrable systems (1.5) admit the so-called pseudop oten tia l represen tation ψ t 2 = A ( ψ t 1 , u ) , ψ t 3 = B ( ψ t 1 , u ) , (1.6) b y means of a pair of equations (1.4 ) whose the compatibility conditions ψ t 2 t 3 = ψ t 3 t 2 are equiv a len t to (1.5 ) . The functions A, B are called pseudop otentials. Suc h a pseudop o ten tial represen tation is a disp ersionless v ersion of the zero curv ature represen tation, whic h is a basic notion in the integrabilit y theory of solitonic equations (see [1]) . One of the in teresting a nd attractive features of the theory of inte grable disp ers ionless equa- tions is that the dep endence of t he pseudop o t entials A ( p, u 1 , . . . , u n ) on p can b e m uc h more 3 complicated then in the solitonic case. F or instance, in [2, 3] some imp ortant examples of pseu- dop oten tials A w ere found related to the Whitham a v eraging pro cedu re for integrable disp ersion PDEs and to the F rob enious manifolds. F or these examples the p -dep endence is determined b y an algebraic curv e of arbitrary gen us g . In the pap er [4] a certain class of pseudop oten tials with mov able singularities w a s describ ed. Some o f the pseudop oten tia ls constructed in [4] are written in terms of degenerate hy p ergeometric f unctions. In the pa p er [5] a wide class of pseudop ot entials A ( p, u 1 , ..., u n ) related to ra tional algebraic curv es w as constructed. These pseudop oten tia ls w ere written in the f o llo wing parametric f orm: A = F 1 ( ξ , u 1 , ..., u n ) , p = F 2 ( ξ , u 1 , ..., u n ) , where t he ξ -dep endence of the functions F i is defined b y the ODE F i,ξ = φ i ( ξ , u 1 , ..., u n ) · ξ − s 1 ( ξ − 1) − s 2 ( ξ − u 1 ) − s 3 ... ( ξ − u n ) − s n +2 . (1.7) Here s 1 , ..., s n +2 are arbitrary constan ts and φ i are p olynomials in ξ of degree n . The dependence of φ i on u 1 , . . . , u n w as describ ed in terms of solutio ns of some o v erdetermined linear system of PDEs with r a tional co efficien ts. In this pap er w e g eneralize this result a nd construct new classes of pseudopotentials A n,k ( p, u 1 , ..., u n ) whose p -dep endence is giv en by (1.7), where φ i ( ξ ) are p o lynomials in ξ of degree n − k , k = 0 , ..., n − 1. W e call the corresponding functions A n,k pseudop otentials of d e- fe ct k . The pseudop otentials of defect 0 are just pseudop oten tials fro m [5] written in a differen t form. W e describe t he pseudop oten tials of defect k in terms of linearly indep enden t solutions o f the follo wing system of linear PDEs with rational co efficien ts ∂ 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, (1.8) 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 (1.9) for o ne unkno wn function h ( u 1 , . . . , u n ). If n = 1, then w e ha v e no equations (1.8) and the single equation (1.9) coincides with the standard hypergeometric equation, u ( u − 1) h ( u ) ′′ + [( α + β + 1) u − γ ] h ( u ) ′ + αβ h ( u ) = 0 , where s 1 = − α, s 2 = α − γ , s 3 = γ − β − 1. Notice, t ha t hypergeometric functions already app eared in connection with disp ersionless PD Es (see, for example [6, 4, 7]). F or arbitrary 4 n the system (1.8), (1.9) can b e solv ed in terms of gener alize d hyp er ge ometric functions (see [8, 9]). Note that the pseudop ot ential A n,k is written in terms of k + 2 linearly indep endent solutions of the system (1.8), (1.9) and therefore the gr o up GL k +2 acts on the set of suc h pseudop otentials. If k = 0 , then this is just the usual a ction of GL 2 on the space of indep enden t v ariables x, t in the equation (1.4). In the case k > 0 the action of larger group GL k +2 is still to b e explained. F or a particular class of 3-dimensional equations the existence of suc h a group of symmetries w as p oin ted out in [10] (see also [11]). It is kno wn [12, 10] that know ledge o f the symmetry group GL n allo ws us to linearize systems of ODEs and PDEs. The pap er is orga nized as follows . In Section 2 we describ e some prop erties of system (1.8), (1.9) and its solutions needed for our purpo ses. Most of these prop erties are w ell know n to exp erts. In Section 3 we rewrite form ulas o f the pap er [5 ] in terms of generalized h yp ergeometric functions. In our pap er the pseudop oten t ials constructed in [5] are called pseudop oten tia ls o f defect 0. A couple of such pseudop oten tials defines a system o f the form ( 1 .5) with l = n. These systems are rewritten in t erms of generalized h yp ergeometric functions in Section 3. W e also pro ve that each of these systems admits n + 1 conserv ation laws of h ydro dyn amic type. In Section 4, for any n and k > 0 we construct pseudop otentials of defe ct k . A couple o f suc h pseudop oten tials defines a system of the fo rm (1.5) with l = n + k . These systems are also constructed in Section 4. The pa r ticular cases n = 3 , k = 1 and n = 5 , k = 3 yield in tegrable equations of the form X i,j P i,j ( z t 1 , z t 2 , z t 3 ) z t i ,t j = 0 , i, j = 1 , 2 , 3 , (1.10) and Q ( z t 1 ,t 1 , z t 1 ,t 2 , z t 1 ,t 3 , z t 2 ,t 2 , z t 2 ,t 3 , z t 3 ,t 3 ) = 0 . (1.11) A classification of all inte grable equations (1.10) and (1.11) w as presen t ed in [11] and in [13], corresp ondingly . Our integrable equations giv e generic solutions of these classification pro blems. In Sections 5 w e construct a nd study a certain class of in tegrable (1+1)-dimensional hy dro- dynamic t yp e systems of the form r i t = v i ( r 1 , ..., r N ) r i x , i = 1 , 2 , ..., N . (1.12) These systems are defined b y an univers al ov erdetermine d compatible system of PDEs of the Gibb ons-Tsarev t yp e [14, 15] for some functions w ( r 1 , ..., r N ), ξ 1 ( r 1 , ..., r N ) , ..., ξ N ( r 1 , ..., r N ) . This system has the follo wing form ∂ i ξ j = ξ j ( ξ j − 1) ξ i − ξ j ∂ i w , ∂ ij w = 2 ξ i ξ j − ξ i − ξ j ( ξ i − ξ j ) 2 ∂ i w ∂ j w , i, j = 1 , ..., N , i 6 = j. (1.13) The only velocities v i ( r 1 , ..., r N ) in (1.12) dep end on n , k . They a r e describ ed b y k + 2 linearly indep enden t solutions of the linear system (1.8), (1.9) (see Section 5 , Theorem 3). One has 5 to substitute functions u 1 = u 1 ( r 1 , ..., r N ) , ..., u n = u n ( r 1 , ..., r N ) for the argumen ts of these solutions. The functions u i are a lso univ ersal. They are defined by the fo llo wing system of PDEs ∂ i u j = u j ( u j − 1 ) ∂ i w ξ i − u j , i = 1 , ..., N , j = 1 , ..., n. (1.14) It is easy to v erify that the system (1 .13), (1.14) is consisten t. Therefore our (1+1)-dimensional systems (1.12) a dmit a lo cal parametrization b y 2 N functions o f one v ariable. F or some v ery sp ecial v alues of parameters s i in (1 .8), (1.9) our systems (1.12) are related to the Whitham hierarchie s [2], to the F robenious manifolds [3, 16], and to t he a sso ciativit y equation [3, 1 6]. In Section 6 w e recall the definition of hy dro dynamic reductions. According to [17 ], the existence of sufficien tly many h ydro dy namic reductions can b e c hosen as a definition of the in tegrability of the systems (1.5 ). W e also recall the definition of integrable pseudopo ten tials (see [7 ]). W e in tro duce the notion of compatible pseudop ot en tials a nd notice that eac h pair of them g iv es a system (1.5) that admits b oth a pseudop oten tia l represen tation and sufficien tly man y h ydro dynamic reductions. W e show that the (1+1) -dimensional hy dro dynamic t yp e systems found in Section 5 are h ydro dynamic reductions of our pseudop otentials A n,k . This implies that these pseudopo ten tials and the corresp onding 3-dimensional systems are in tegrable in t he sense of the definitions men tio ned ab ov e (see Theorem 4). 2 Generalize d h yp ergeometric funct ions The follo wing statemen ts can b e v erified straigh tf o rw ardly . Prop osition 1. The system of linear equations (1.8), ( 1.9) is compatible for a n y constan ts s 1 , . . . , s n +2 . The dimension of the linear space H of solutions of the system (1.8), (1.9) is equal to n + 1.  W e call elemen ts of H gener alize d hyp er ge ometric functions. Prop osition 2. The system (1.8), (1.9) is equiv alent to the follo wing system Q i ( u 1 ∂ ∂ u 1 , ..., u n ∂ ∂ u n ) u − 1 i h = P i ( u 1 ∂ ∂ u 1 , ..., u n ∂ ∂ u n ) h, i = 1 , ..., n (2.15) where Q i ( k 1 , ..., k n ) = ( k 1 + ... + k n − s 1 − ... − s n +1 )( k i + 1) , P i ( k 1 , ..., k n ) = ( k 1 + ... + k n − 1 − s 1 − ... − s n +2 )( k i − s i ) .  Recall that a system o f the form (2.1 5) is called a hypergeometric system [9]. It can b e solv ed in terms of the so-called Horn series [9 ]. 6 Example 1. The system (2.15) and hence (1.8 ), (1.9) has a unique solution ho lo morphic at the p oint 0 = ( 0 , . . . , 0) suc h that h ( 0 ) = 1 . The deriv ativ es of this solution at 0 is give n by h ( k 1 ,...,k n ) ( 0 ) = Q k 1 + ... + k n − 1 j =0 (1 − j + s n +2 + r ) Q k 1 + ... + k n − 1 j =0 ( − j + r ) n Y j =1 k j − 1 Y i =0 ( i − s j ) , where r = n +1 X i =1 s i . Let us denote the solution describ ed in this example by F ( s 1 , . . . , s n +2 , u 1 , . . . , u n ). F or brevit y , we a lso will use the notatio n F ( s 1 , . . . , s n +2 ). Prop osition 3. The function F ( s 1 , . . . , s n +2 ) admits the follo wing integral represen ta tion F ( s 1 , . . . , s n +2 , u 1 , . . . , u n ) = C Z 1 0 t − 2 − r − s n +2 (1 − t ) s n +2 (1 − tu 1 ) s 1 · · · (1 − tu n ) s n dt, where C = Γ( − r ) Γ(1 + s n +2 )Γ( − 1 − r − s n +2 ) .  It is w ell-kno wn tha t for the standard h yp ergeometric equation there exist the Laplace transformations shifting t he parameters by 1 . Analogies o f suc h transformations f o r the system (1.8), (1.9) are given b y Prop osition 4. The following iden t it ies hold: ∂ F ( s 1 , . . . , s i , . . . s n +2 ) ∂ u i = − s i (1 + r + s n +2 ) r F ( s 1 , . . . , s i − 1 , . . . s n +2 ) , i ≤ n, L 1  F ( s 1 , . . . , s n +1 , s n +2 )  = s n +1 (1 + r + s n +2 ) r F ( s 1 , . . . , s n +1 − 1 , s n +2 ) , L 2  F ( s 1 , . . . , s n +1 , s n +2 )  = (1 + r + s n +2 ) F ( s 1 , . . . , s n +1 , s n +2 − 1 ) , where L 1 = n X j =1 (1 − u j ) ∂ ∂ u j + (1 + r + s n +2 ) , L 2 = − n X j =1 u j ∂ ∂ u j + (1 + r + s n +2 ) , and M i  F ( s 1 , . . . , s i , . . . s n +2 )  = (1 + r ) F ( s 1 , . . . , s i + 1 , . . . s n +2 ) , i ≤ n, M n +1  F ( s 1 , . . . , s n +1 , s n +2 )  = (1 + r ) F ( s 1 , . . . , s n +1 + 1 , s n +2 ) , 7 M n +2  F ( s 1 , . . . , s n +1 , s n +2 )  = − (1 + s n +2 ) F ( s 1 , . . . , s n +1 , s n +2 + 1) , where M i = n X j =1 u j ( u j − 1) ∂ ∂ u j − n X j =1 s j u j − (2 + r + s n +2 ) u i + (1 + r ) , i ≤ n, M n +1 = n X j =1 u j ( u j − 1) ∂ ∂ u j − n X j =1 s j u j + (1 + r ) , M n +2 = n X j =1 u j ( u j − 1) ∂ ∂ u j − n X j =1 s j u j − ( 1 + s n +2 ) . F urthermore, let H s 1 ,...,s n +2 b e the space o f solutions of the system ( 1 .8), (1.9). W e ha ve ∂ ∂ u i H s 1 ,...,s n +2 ⊂ H s 1 ,...,s i − 1 ,...,s n +2 , L 1 H s 1 ,...,s n +2 ⊂ H s 1 ,...,s n +1 − 1 ,s n +2 , L 2 H s 1 ,...,s n +2 ⊂ H s 1 ,...,S n +2 − 1 , M i H s 1 ,...,s n +2 ⊂ H s 1 ,...,s i +1 ,...,s n +2 .  Prop osition 5. Let H = H s 1 ,...,s n +2 and e H = H s 1 ,...,s n , 0 ,s n +1 ,s n +2 . Then e H is spanned b y H and the function Z ( u 1 , ..., u n , u n +1 ) = Z u n +1 0 ( t − u 1 ) s 1 · · · ( t − u n ) s n t s n +1 ( t − 1) s n +2 dt. (2.16) Moreo ve r, the space H s 1 ,...,s n , 0 ,..., 0 ,s n +1 ,s n +2 ( m zeros) is spanned b y H and Z ( u 1 , ..., u n , u n +1 ), Z ( u 1 , ..., u n , u n +2 ) , ..., Z ( u 1 , ..., u n , u n + m ).  3 Pseudop oten t ials o f defec t 0 Most results of this Section w a s obtained in a differen t for m in the pap er [5]. F or an y generalized hy p ergeometric f unction g ∈ H w e put S n ( g , ξ ) = X 1 ≤ i ≤ n u i ( u i − 1)( ξ − u 1 ) ... ˆ i... ( ξ − u n ) g u i + (1 + X 1 ≤ i ≤ n +2 s i )( ξ − u 1 ) ... ( ξ − u n ) g . (3.17) Here g u i = ∂ g ∂ u i . It is clear that S n ( g , ξ ) is a p olynomial of degree n in ξ . Example 2. In the simplest case n = 1 w e hav e S 1 ( g , ξ ) = u ( u − 1) g u + (1 + s 1 + s 2 + s 3 )( ξ − u ) g 8 where u = u 1 . W e need the follo wing prop erty o f the p olynomial S n ( g , ξ ): Lemma 1. F or any 1 ≤ m ≤ n the follo wing iden tity is v alid u m ( u m − 1 )( u m − u 1 ) ... ˆ m... ( u m − u n ) S n ( g , ξ ) u m + ( s m +1) S n ( g, ξ ) ξ − u m S n ( g , u m ) = ξ ( ξ − 1)( ξ − u 1 ) ... ˆ m... ( ξ − u n )  s 1 ξ − u 1 + ... + s m + 1 ξ − u m + ... + s n ξ − u n + s n +1 ξ + s n +2 ξ − 1  .  Define P n ( g , ξ ) by the form ula P n ( g , ξ ) = Z ξ 0 S n ( g , ξ )( ξ − u 1 ) − s 1 − 1 ... ( ξ − u n ) − s n − 1 ξ − s n +1 − 1 ( ξ − 1) − s n +2 − 1 dξ (3.18) if Re s n +1 < − 1 and as the analytic con tin uatio n of this express ion otherwise. Prop osition 6. The expression P n ( g , ξ ) u m S n ( g , u m ) (3.19) do es not depend on g . More precisely , u m ( u m − 1 )( u m − u 1 ) ... ˆ m... ( u m − u n ) P n ( g , ξ ) u m S n ( g , u m ) = − ( ξ − u 1 ) − s 1 ... ( ξ − u m ) − s m − 1 ... ( ξ − u n ) − s n ξ − s n +1 ( ξ − 1) − s n +2 . (3.20) Pro of. The deriv ativ e of (3.19) with resp ect to ξ is equal to S n ( g , ξ ) u m + ( s m +1) S n ( g, ξ ) ξ − u m S n ( g , u m ) ( ξ − u 1 ) − s 1 − 1 ... ( ξ − u n ) − s n − 1 ξ − s n +1 − 1 ( ξ − 1 ) − s n +2 − 1 . Lemma 1 implies that this deriv ativ e do es not dep end on g . Since the v alue o f (3.19) at ξ = 0 is equal to zero, expression (3.19) it self do es not dep end of g . Iden tity (3 .20) also fo llo ws from Lemma 1.  Let g 1 , g 0 b e linearly indep enden t elemen ts of H . A pseudop oten t ia l A n ( p, u 1 , ..., u n ) defined in a parametric form b y A n = P n ( g 1 , ξ ) , p = P n ( g 0 , ξ ) (3.21) is called p seudop otential of defe ct 0 . Relations (3.21) mean that to find A n ( p, u 1 , ..., u n ) , w e hav e to express ξ from the second equation and substitute the result into the first equation. 9 Let g 0 , g 1 , ..., g n ∈ H b e a basis in H . D efine pseudop oten tials B α ( p, u 1 , ..., u n ) of defect 0, where α = 1 , ..., n, b y B α = P n ( g α , ξ ) , p = P n ( g 0 , ξ ) , α = 1 , ..., n. Supp ose that u 1 , ..., u n are functions of t 0 = x, t 1 , ..., t n . Theorem 1. The compatibility conditions ψ t α t β = ψ t β t α for the sys tem ψ t α = B α ( ψ x , u 1 , ..., u n ) , α = 1 , ..., n. (3.22) are equiv alen t to the fo llo wing system of PDEs for u 1 , ..., u n : 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 + 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 + (3.23 ) 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 . Here q , r, s run f r om 0 to n and t 0 = x . Pro of. If B α are giv en in a parametric form B α = f α ( ξ , u 1 , ..., u n ) , p = f 0 ( ξ , u 1 , ..., u n ) , then the compatibilit y conditions fo r (3.22) is equiv alen t to n X i =1  ( f q ,ξ f r,u i − f r,ξ f q ,u i ) u i,t s + ( f r,ξ f s,u i − f s,ξ f r,u i ) u i,t q + ( f s,ξ f q ,u i − f q ,ξ f s,u i ) u i,t r  = 0 . (3.24 ) T aking in to account (3.1 8 ), (3.20), w e get f q ,ξ f r,u i − f r,ξ f q ,u i =  S n ( g q , ξ ) P n ( g r , ξ ) u i − S n ( g r , ξ ) P n ( g q , ξ ) u i  ( ξ − u 1 ) − s 1 − 1 · · · ( ξ − u n ) − s n − 1 ξ − s n +1 − 1 ( ξ − 1) − s n +2 − 1 = S n ( g q , ξ ) S n ( g r , u i ) − S n ( g r , ξ ) S n ( g q , u i ) ( ξ − u i ) · u i ( u i − 1 )( u i − u 1 ) ... ˆ i... ( u i − u n ) · T = S n ( g q , ξ ) g r,u i − S n ( g r , ξ ) g q ,u i ξ − u i · T . Here T = − ( ξ − u 1 ) − 2 s 1 − 1 ... ( ξ − u n ) − 2 s n − 1 ξ − 2 s n +1 − 1 ( ξ − 1) − 2 s n +2 − 1 10 do es not dep end on i . Using the ab ov e f orm ula for f q ,ξ f r,u i − f r,ξ f q ,u i and similar form ulas fo r f r,ξ f s,u i − f s,ξ f r,u i , f s,ξ f q ,u i − f q ,ξ f s,u i , w e can rewrite (3.24) as follow s: X 1 ≤ i ≤ n  S n ( g q , ξ ) g r,u i − S n ( g r , ξ ) g q ,u i ξ − u i u i,t s + S n ( g r , ξ ) g s,u i − S n ( g s , ξ ) g r,u i ξ − u i u i,t q + S n ( g s , ξ ) g q ,u i − S n ( g q , ξ ) g s,u i ξ − u i u i,t r  = 0 . It f ollo ws from (3.17) that t he left hand side is a p o lynomial o f degree n − 1 in ξ . T o conclude the pro of, it remains to ev aluate this p olynomial at ξ = u 1 , ..., u n .  Remark 1. Given t 1 , t 2 , t 3 , Theorem 1 yields a 3- dimensional system of the fo rm (1 .5 ) with l = n equations p osse ssing pseudop oten tial represen tation. Remark 2. A system of PDEs for u 1 , ..., u n , whic h is equiv a len t to compatibilit y conditions for equations of the form (3.24), was called in [2] a Whitham hier ar chy . In the pap er [2] I.M. Kric hev er constructed some Whitham hierarc hies related to algebraic curv es of arbitrary gen us g . The hierarc hy corr esp o nding to g = 0 is equiv alen t to one describ ed b y Theorem 1 if s 1 = . . . = s n +2 = 0 . In this case the vec tor space H is spanned b y 1 , u 1 , u 2 , . . . , u n . Prop osition 7. The system (3.23) p ossesses n + 1 h ydro dynamic type conserv ation la ws. Pro of. Let b H = H − 2 s 1 ,..., − 2 s n , − 2 s n +1 − 1 , − 2 s n +2 − 1 b e the space of generalized h yp ergeometric functions defined b y (1.8), (1.9) with ˆ s i = − 2 s i for i = 1 , ..., n and ˆ s i = − 2 s i − 1 f o r i = n + 1 , n + 2 . Let Z ∈ b H b e an a r bitrary elemen t in b H . Denote b y X j the left hand side of (3.23). Define functions A i , B i , C i b y n X i =1 ( A i u i,t q + B i u i,t r + C i u i,t s ) = n X j =1 1 s j Z u j X j . One can c hec k that ( A i ) u j = ( A j ) u i , ( B i ) u j = ( B j ) u i , ( C i ) u j = ( C j ) u i . Therefore A i = A u i , B i = B u i , C i = C u i for some functions A, B , C and w e ha ve A t q + B t r + C t s = 0 . Since dim b H = n + 1 , we obtain n + 1 conserv ation law s of the h ydro dynamic type. 4 Pseudop oten t ials o f defec t k > 0 In t his section w e construct a new class of pseudop otentials. W e call them pseudop otentials of defe ct k . T o define pseudop oten tials of defect k , w e fix k linearly indep enden t generalized h yp ergeometric functions h 1 , ..., h k ∈ H . F or an y g ∈ H define S n,k ( g , ξ ) by the form ula S n,k ( g , ξ ) = 1 ∆ X 1 ≤ i ≤ n − k +1 u i ( u i − 1 )( ξ − u 1 ) ... ˆ i... ( ξ − u n − k +1 )∆ i ( g ) . (4.25) 11 Here ∆ = det     h 1 ... h k h 1 ,u n − k +2 ... h k ,u n − k +2 ......... ... ......... h 1 ,u n ... h k ,u n     , ∆ i ( g ) = det       g h 1 ... h k g u i h 1 ,u i ... h k ,u i g u n − k +2 h 1 ,u n − k +2 ... h k ,u n − k +2 ......... ... ... ......... g u n h 1 ,u n ... h k ,u n       . It is clear that S n,k ( g , ξ ) is a p olynomial in ξ of degree n − k . Notice that S n,k ( h 1 , ξ ) = ... = S n,k ( h k , ξ ) = 0. It is easy to see that linear tra nsformations h i → c i 1 h 1 + ... + c ik h k , g → g + d 1 h 1 + ... + d k h k with constan t co effic ien ts c ij , d i do not c hange S n,k ( g , ξ ). Example 3. In the simplest case n = 2 , k = 1 we hav e S 2 , 1 ( g , ξ ) = u 1 ( u 1 − 1 )( ξ − u 2 ) g h 1 ,u 1 − g u 1 h 1 h 1 + u 2 ( u 2 − 1 )( ξ − u 1 ) g h 1 ,u 2 − g u 2 h 1 h 1 . Lemma 2. If 1 ≤ m < n − k + 2 , then the following identit y is v alid: u m ( u m − 1 )( u m − u 1 ) ... ˆ m... ( u m − u n ) S n,k ( g , ξ ) u m + ( s m +1) S n,k ( g, ξ ) ξ − u m S n,k ( g , u m ) = − ( u m − u n − k +2 ) ... ( u m − u n ) 1 ∆ X 1 ≤ i ≤ n − k +1 u i ( u i − 1 )( ξ − u 1 ) ... ˆ i... ( ξ − u n − k +1 ) f ∆ i + 1 ∆ X n − k +2 ≤ i ≤ n, 1 ≤ j ≤ n − k +1 ( u m − u n − k +2 ) ... ˆ i... ( u m − u n ) s i u j ( u j − 1 )( ξ − u 1 ) ... ˆ j ... ( ξ − u n − k +1 ) g ∆ i,j + ( s m + 1) u m ( u m − 1 )( u m − u n − k +2 ) ... ( u m − u n )( ξ − u 1 ) ... ˆ m... ( ξ − u n − k +1 ) ξ − u m + ( u m − u n − k +2 ) ... ( u m − u n ) X 1 ≤ i ≤ n − k +1 ,i 6 = m s i u i ( u i − 1 ) Y 1 ≤ j ≤ n − k +1 ,j 6 = i,m ( ξ − u j )+ ( u m − u n − k +2 ) ... ( u m − u n )( ξ − u 1 ) ... ˆ m... ( ξ − u n − k +1 )( X 1 ≤ i ≤ n − k +1 ( u m + u i − 1 ) s i + 2 u m − 1 )+ u m ( u m − 1 )( ξ − u 1 ) ... ˆ m... ( ξ − u n − k +1 ) X n − k +2 ≤ i ≤ n ( u m − u n − k +2 ) ... ˆ i... ( u m − u n ) s i + ( u m − u n − k +2 ) ... ( u m − u n )( ξ − u 1 ) ... ˆ m... ( ξ − u n − k +1 )(( u m − 1) s n +1 + u m s n +2 ) . If n − k + 2 ≤ m, then u m ( u m − 1 )( u m − u 1 ) ... ˆ m... ( u m − u n ) S n,k ( g )( ξ ) u m + s m S n,k ( g, ξ ) ξ − u m S n,k ( g , u m ) = 1 ∆ X n − k +2 ≤ i ≤ n, 1 ≤ j ≤ n − k +1 ( u m − u n − k +2 ) ... ˆ i... ( u m − u n ) s i u j ( u j − 1 )( ξ − u 1 ) ... ˆ j ... ( ξ − u n − k +1 ) g ∆ i,j + 12 s m u m ( u m − 1 )( u m − u n − k +2 ) ... ˆ m... ( u m − u n )( ξ − u 1 ) ... ( ξ − u n − k +1 ) ξ − u m . Here f ∆ i = det     h 1 ,u i ... h k ,u i h 1 ,u n − k +2 ... h k ,u n − k +2 ......... ... ......... h 1 ,u n ... h k ,u n     and g ∆ i,j is obtained from ∆ b y replacing the ro w ( h 1 ,u i , ..., h k ,u i ) b y ( h 1 ,u j , ..., h k ,u j ).  Define functions P n,k ( g , ξ ) b y P n,k ( g , ξ ) = (4.26) Z ξ 0 S n,k ( g , ξ )( ξ − u 1 ) − s 1 − 1 ... ( ξ − u n − k +1 ) − s n − k +1 − 1 ( ξ − u n − k +2 ) − s n − k +2 ... ( ξ − u n ) − s n ξ − s n +1 − 1 ( ξ − 1) − s n +2 − 1 dξ if Re s n +1 < − 1 , and as the analytic con tin uatio n of this express ion otherwise. Prop osition 8. The expression P n,k ( g , ξ ) u m S n,k ( g , u m ) (4.27) do es not depend on g . Moreo ve r, w e ha v e X 1 ≤ i ≤ k +1 u m i ( u m i − 1)( u m i − u 1 ) ... \ m 1 , ..., m k +1 ... ( u m i − u n ) P n,k ( g , ξ ) u m i S n,k ( g , u m i ) = (4.28) − ( ξ − u 1 ) − s 1 ... ( ξ − u n − k +1 ) − s n − k +1 ( ξ − u n − k +2 ) − s n − k +2 +1 ... ( ξ − u n ) − s n +1 ξ − s n +1 ( ξ − 1) − s n +2 ( ξ − u m 1 ) ... ( ξ − u m k +1 ) . Pro of. The deriv ativ e of expres sion (4 .2 7) with r esp ect to ξ is equal to S n,k ( g , ξ ) u m + ( s m +1) S n,k ( g, ξ ) ξ − u m S n,k ( g , u m ) ( ξ − u 1 ) − s 1 − 1 · · · ( ξ − u n − k +1 ) − s n − k +1 − 1 ( ξ − u n − k +2 ) − s n − k +2 ... ( ξ − u n ) − s n ξ − s n +1 − 1 ( ξ − 1) − s n +2 − 1 for 1 ≤ m < n − k + 2 and is equal to S n,k ( g , ξ ) u m + s m S n,k ( g, ξ ) ξ − u m S n,k ( g , u m ) ( ξ − u 1 ) − s 1 − 1 · · · ( ξ − u n − k +1 ) − s n − k +1 − 1 ( ξ − u n − k +2 ) − s n − k +2 ... ( ξ − u n ) − s n ξ − s n +1 − 1 ( ξ − 1) − s n +2 − 1 otherwise. Lemma 2 implies that this deriv a t ive do es not dep end on g . Moreov er, the v a lue of the expression (4.27) at ξ = 0 is equal to zero. Therefore the expression (4 .2 7) itself do es not dep end on g . The pro o f of (4.28) is similar.  13 Let g 1 , g 2 ∈ H . Assume that g 1 , g 2 , h 1 , ..., h k are linearly indep enden t. Define pseudop oten- tial A n,k ( p, u 1 , ..., u n ) in pa r ametric form b y A n,k = P n,k ( g 1 , ξ ) , p = P n,k ( g 2 , ξ ) . (4.29) T o construct A n,k ( p, u 1 , ..., u n ), w e find ξ from the second equation and substitute into the first one. The pseudopo t ential A n,k ( p, u 1 , ..., u n ) is called p seudop otential of defe ct k . Theorem 2. Let g 0 , g 1 , ..., g n − k , h 1 , ..., h k ∈ H b e a basis in H and B α , α = 1 , ..., n − k ar e defined b y B α = P n,k ( g α , ξ ) , p = P n,k ( g 0 , ξ ) , α = 1 , ..., n − k . Then the compatibility conditions f or (3.22) a re equiv alen t to the following system of PDEs for u 1 , ..., u n : X 1 ≤ i ≤ n − k ,i 6 = j  ∆ j ( g q )∆ i ( g r ) − ∆ j ( g r )∆ i ( g q )  u j ( u j − 1) u i,t s − u i ( u i − 1 ) u j,t s u j − u i + X 1 ≤ i ≤ n − k ,i 6 = j  ∆ j ( g r )∆ i ( g s ) − ∆ j ( g s )∆ i ( g r )  u j ( u j − 1) u i,t q − u i ( u i − 1) u j,t q u j − u i + (4.30) X 1 ≤ i ≤ n − k ,i 6 = j  ∆ j ( g s )∆ i ( g q ) − ∆ j ( g q )∆ i ( g s )  u j ( u j − 1 ) u i,t r − u i ( u i − 1 ) u j,t r u j − u i = 0 , where j = 1 , ..., n − k and n − k +1 X i =1 ∆ i ( g r ) u i,t s = n − k +1 X i =1 ∆ i ( g s ) u i,t r , (4.31) n − k +1 X i =1 ∆ i ( g r ) u m ( u m − 1 ) u i,t s − u i ( u i − 1 ) u m,t s u m − u i = n − k +1 X i =1 ∆ i ( g s ) u m ( u m − 1 ) u i,t r − u i ( u i − 1 ) u m,t r u m − u i , (4.32) where m = n − k + 2 , ..., n . Here q , r , s run from 0 to n and t 0 = x . Pro of. W e ha ve to explicitly calculate the co efficien ts in (3.2 4). Using (4.26), (4.2 7), w e find that f q ,ξ f r,u i − f r,ξ f q ,u i =  S n,k ( g q , ξ ) P n,k ( g r , ξ ) u i − S n,k ( g r , ξ ) P n,k ( g q , ξ ) u i  · T =  S n,k ( g q , ξ ) S n,k ( g r , u i ) − S n,k ( g r , ξ ) S n,k ( g q , u i )  · P n,k ( g q , ξ ) u i S n,k ( g q , u i )) · T . Similar form ula s are v alid for f r,ξ f s,u i − f s,ξ f r,u i , f s,ξ f q ,u i − f q ,ξ f s,u i . Here T = ( ξ − u 1 ) − s 1 − 1 ... ( ξ − u n − k +1 ) − s n − k +1 − 1 ( ξ − u n − k +2 ) − s n − k +2 ... ( ξ − u n ) − s n ξ − s n +1 − 1 ( ξ − 1) − s n +2 − 1 14 do es not dep end on i . Using (4.28), w e can express P n,k ( g q , ξ ) u i S n,k ( g q , u i ) , i = 1 , ..., n − k in terms of P n,k ( g q , ξ ) u m S n,k ( g q , u m ) , m = n − k + 1 , ..., n, whic h are linearly indep enden t a s functions of ξ . Substituting these into (3.24), w e obtain n − k X i =1  S n,k ( g q , ξ ) S n,k ( g r , u i ) − S n,k ( g r , ξ ) S n,k ( g q , u i ) ( ξ − u i ) · u i ( u i − 1 )( u i − u 1 ) ... ˆ i... ( u i − u n − k ) u i,t s + S n,k ( g r , ξ ) S n,k ( g s , u i ) − S n,k ( g s , ξ ) S n,k ( g r , u i ) ( ξ − u i ) · u i ( u i − 1 )( u i − u 1 ) ... ˆ i... ( u i − u n − k ) u i,t q + S n,k ( g s , ξ ) S n,k ( g q , u i ) − S n,k ( g q , ξ ) S n,k ( g s , u i ) ( ξ − u i ) · u i ( u i − 1 )( u i − u 1 ) ... ˆ i... ( u i − u n − k ) u i,t r  = 0 , (4.33) n − k X i =1  S n,k ( g q , ξ ) S n,k ( g r , u i ) − S n,k ( g r , ξ ) S n,k ( g q , u i ) ( u i − u m ) · u i ( u i − 1 )( u i − u 1 ) ... ˆ i... ( u i − u n − k ) u i,t s + S n,k ( g r , ξ ) S n,k ( g s , u i ) − S n,k ( g s , ξ ) S n,k ( g r , u i ) ( u i − u m ) · u i ( u i − 1)( u i − u 1 ) ... ˆ i... ( u i − u n − k ) u i,t q + S n,k ( g s , ξ ) S n,k ( g q , u i ) − S n,k ( g q , ξ ) S n,k ( g s , u i ) ( u i − u m ) · u i ( u i − 1)( u i − u 1 ) ... ˆ i... ( u i − u n − k ) u i,t r  + S n,k ( g q , ξ ) S n,k ( g r , u m ) − S n,k ( g r , ξ ) S n,k ( g q , u m ) u m ( u m − 1)( u m − u 1 ) ... ( u m − u n − k ) u m,t s + S n,k ( g r , ξ ) S n,k ( g s , u m ) − S n,k ( g s , ξ ) S n,k ( g r , u m ) u m ( u m − 1 )( u m − u 1 ) ... ( u m − u n − k ) u m,t q + S n,k ( g s , ξ ) S n,k ( g q , u m ) − S n,k ( g q , ξ ) S n,k ( g s , u m ) u m ( u m − 1 )( u m − u 1 ) ... ( u m − u n − k ) u m,t r = 0 , (4.34) where m = n − k + 1 , ..., n . One can che c k straightforw ardly that (4.34) is equiv alen t to (4 .31) for m = n − k + 1 and to ( 4 .32) for m = n − k + 2 , ..., n . Notice that the left hand side of equation (4 .33) is a p olynomial in ξ of degree n − k − 1. Ev aluat ing this p olynomial at ξ = u j , j = 1 , ..., n − k w e obtain (4.3 0).  Remark 3. Given t 1 , t 2 , t 3 , Theorem 2 yields a 3- dimensional system of the fo rm (1 .5 ) with l = n + k equations p oss essing pseudop otential represen tation. Indeed, the for m ulas (4 .31), (4.32) giv e 3 k linearly indep enden t equations if q , r , s = 1 , 2 , 3 . The formula (4.3 0) gives n − k equations. On the other hand, one can construct exactly k linear combinations o f equations (4.31), (4.3 2) with q , r, s = 1 , 2 , 3 suc h that deriv ativ es of u i , i = n − k + 1 , ..., n cancel out. Moreo ve r, these linear comb inations b elong to the span of equations (4.30). Therefore, there exist ( n − k ) + 3 k − k = n + k linearly indep enden t equations. Remark 4. In (4 .30), (4.31), (4 .32) we hav e to assume n ≥ k + 2. Indeed, for n = k + 1 w e cannot construct more then one pseudop ot en tial and therefore there is no an y system of 15 the f o rm (1.5) asso ciated with this case. How ev er the corresp onding pseudop otential generates in teresting integrable (1 +1)-dimensional systems of hydrodynamic t yp e (see Section 5). Prob- ably t hese pseudopo t entials for k = 0 , 1 , ... a re also related to some infinite in tegra ble c hains of the Benney t yp e [18, 19]. The system (4.30)-(4 .32) p o ssess es many conserv ation laws of the hy dro dynamic type. In particular, the follo wing statement can b e ve rified by a straigh tforw a r d calculation. Prop osition 9. F or any r 6 = s = 0 , 1 , ..., n there exist k conserv at io n laws fo r the system (4.30)-(4.32) of the fo rm: ∆( g r , h 1 , ... ˆ i...h k ) ∆( h 1 , ..., h k ) ! t s = ∆( g s , h 1 , ... ˆ i...h k ) ∆( h 1 , ..., h k ) ! t r , (4.35) where i = 1 , ..., k . Here ∆( f 1 , ..., f k ) = det     f 1 ... f k h 1 ,u n − k +2 ... f k ,u n − k +2 ......... ... ......... f 1 ,u n ... f k ,u n     . Prop osition 9 allo ws us to define functions z 1 , ..., z k suc h that ∆( g r , h 1 , ... ˆ i...h k ) ∆( h 1 , ..., h k ) = z i,t r (4.36) for all i = 1 , ..., k and r = 0 , 1 , ..., n . Supp ose n ≥ 3 k ; then the system of the form (1.5) o btained from (4.3 0), (4 .31), ( 4 .32) with q , r, s = 1 , 2 , 3 consists of 3 k equations (4.31), (4.32) (they are equiv alen t to (4.35)) and n − 2 k equations of the fo rm (4.30). Indeed, only n − 2 k equations (4.30) are linearly indep enden t on (4.31), (4.32 ) . Expressing u 1 , ..., u 3 k in terms of z i,t 1 , z i,t 2 , z i,t 3 , i = 1 , ..., k from (4.3 6) and substituting in to n − 2 k equations of the form (4.3 0), w e obta in a 3-dimensional system of n − 2 k equations for n − 2 k unkno wns z 1 , ..., z k , u 3 k + 1 , ...u n . This is a quasi-linear system o f the second order with resp ect to z i and of the first order with resp ect to u j , whose co effic ien ts dep end on z i,t 1 , z i,t 2 , z i,t 3 , i = 1 , ..., k and u 3 k + 1 , ...u n . It is clear that the general solution of t he system can be lo cally parameterized by n − k functions in t wo v ariables. In t he case 2 k ≤ n < 3 k the functions z i,t 1 , z i,t 2 , z i,t 3 , i = 1 , ..., k are functionally dep enden t. W e ha ve 3 k − n equations of the form R i ( z 1 ,t 1 , z 1 ,t 2 , z 1 ,t 3 , ..., z k ,t 1 , z k ,t 2 , z k ,t 3 ) = 0 , i = 1 , ..., 3 k − n and n − 2 k second order quasi-linear equations. T ota lly w e ha v e (3 k − n ) + ( n − 2 k ) = k equations for k unkno wns z 1 , ..., z k . It is clear tha t t he general solution of this system can b e lo cally parameterized b y n − k functions in t w o v ariables. 16 Supp ose n < 2 k ; then w e ha ve n + k < 3 k , which means that 3 k equations of the form (4.31), ( 4 .32) are linearly dep endent. Probably in this case the general solution of the system can a lso b e lo cally parameterized by n − k functions in t wo v a riables. One of the most interes ting cases is n = 3 k , when w e ha v e a system of k quasi-linear second order equations for the functions z 1 , ..., z k . Consider the simplest case k = 1 . Example 4. In t he case n = 3 , k = 1 the fo rm ulas (4.30), (4 .31) can b e rewritten as follo ws. L et h 1 , g 0 , g 1 , g 2 b e linearly indep en den t elemen ts of H . Denote by B ij the cof a ctors of the matrix     h 1 g 0 g 1 g 2 h 1 ,u 1 g 0 ,u 1 g 1 ,u 1 g 1 ,u 1 h 1 ,u 2 g 0 ,u 2 g 1 ,u 2 g 1 ,u 1 h 1 ,u 3 g 0 ,u 3 g 1 ,u 3 g 1 ,u 3     . Define v ector fields V i b y V 1 = B 22 ∂ ∂ t 0 + B 23 ∂ ∂ t 1 + B 24 ∂ ∂ t 2 , V 2 = B 32 ∂ ∂ t 0 + B 33 ∂ ∂ t 1 + B 34 ∂ ∂ t 2 , V 3 = B 42 ∂ ∂ t 0 + B 43 ∂ ∂ t 1 + B 44 ∂ ∂ t 2 . Then ( 4.31) is equiv alent to V 1 ( u 2 ) = V 2 ( u 1 ) , V 2 ( u 3 ) = V 3 ( u 2 ) , V 3 ( u 1 ) = V 1 ( u 3 ) . (4.37) Relation (4.30) leads to one more equation u 3 ( u 3 − 1)( u 1 − u 2 ) V 1 ( u 2 ) + u 1 ( u 1 − 1)( u 2 − u 3 ) V 2 ( u 3 ) + u 2 ( u 2 − 1)( u 3 − u 1 ) V 3 ( u 1 ) = 0 . (4.3 8 ) The conserv ation la ws (4.35) hav e the form  g r h 1  t s =  g s h 1  t r . In tro ducing z suc h t hat z t r = g r h 1 , we reduce (4.38) to a quasi-linear equation of the form X i,j P i,j ( z t 0 , z t 1 , z t 2 ) z t i ,t j = 0 , i, j = 0 , 1 , 2 . (4.39) In the pap er [11] an inexplicit description of all integrable equations (4 .39) was prop osed. The equation constructed ab o v e corresp onds to the generic case in this classification. I ndeed, it dep ends on fiv e essen tial parameters s 1 , ..., s 5 whic h ag r ees with the results of [11]. F or in teger v alues of parameters s i equations (1.8), (1.9) can b e solv ed in elemen tary func- tions. This prov ides simple examples of equations (4.39) ha ving pseudopo t en tials. In the mo st 17 degenerate case s 1 = · · · = s 5 = 0 one can c ho ose h 1 = 1 , g 0 = u 1 , g 1 = u 2 , g 2 = u 3 . The corresp onding equation (4.39) is giv en by z t 2 ( z t 2 − 1 )( z t 0 − z t 1 ) z t 0 t 1 + z t 0 ( z t 0 − 1 )( z t 1 − z t 2 ) z t 1 t 2 + z t 1 ( z t 1 − 1 )( z t 2 − z t 0 ) z t 2 t 0 = 0 . More general examples o f equations P 1 ( z t 0 , z t 1 , z t 2 ) z t 0 t 1 + P 2 ( z t 0 , z t 1 , z t 2 ) z t 1 t 2 + P 3 ( z t 0 , z t 1 , z t 2 ) z t 2 t 0 = 0 (4.40) corresp ond to s 1 = s 2 = s 3 = 0. In this case one can choose h = 1, g 0 = f ( u 1 ) , g 1 = f ( u 2 ) , g 2 = f ( u 3 ), where f ′ ( x ) = x s 4 ( x − 1) s 5 . In the new v ariables ¯ u i = f ( u i ) the system (4.37), (4.38) is equiv alen t to a single equation of the form (4.40). One of the results of the pap er [11] is a complete classification of equations (4.4 0) p ossessing a pseudop oten tia l represen tation. The ab o ve example seems to b e the generic case in this classification.  The sys tem (4.3 0)-(4.32) has conserv at io n la ws different from (4 .3 5). Conjecture. The system (4.30) - (4.3 2) p osse sses n + 1 conserv ation law s of the general form A t q + B t r + C t s = 0 additional to (4.3 5). This family of conserv at io n la ws can b e parameterized b y elemen t s from b H = H − 2 s 1 ,..., − 2 s n , − 2 s n +1 − 1 , − 2 s n +2 − 1 (cf. Prop osition 7). This conjecture is supp orted by some computer computations for small n and k . Remark 5. Let us mak e in (4.26) a c hange of v ariables of the form ξ → aξ + b cξ + d , u 1 → φ 1 , ..., u n → φ n , (4.41) where a, b, c, d , φ 1 , ..., φ n are arbitrary functions in u 1 , ..., u n . After that w e get under the in tegral in (4.26) an expression of the form S ( ξ )( ξ − ρ 1 ) − s 1 − 1 ... ( ξ − ρ n − k +1 ) − s n − k +1 − 1 ( ξ − ρ n − k +2 ) − s n − k +2 ... ( ξ − ρ n ) − s n ( ξ − ρ n +1 ) − s n +1 − 1 × ( ξ − ρ n +2 ) − s n +2 − 1 ( ξ − ρ n +3 ) s 1 + ... + s n +2 +1 , where S ( ξ ) is a p o lynomial in ξ of degree n − k and ρ 1 , ..., ρ n +3 are functions of u 1 , ..., u n . Therefore the num b ers {− s 1 − 1 , ..., − s n − k +1 − 1 , − s n − k +2 , ..., − s n , − s n +1 − 1 , − s n +2 − 1 , s 1 + ... + s n +2 + 1 } (4.42) pla y a symmetric ro le in the constructed pseudop ot entials A n,k . Using transformat ions (4.41), one can c ho ose any three of the functions ρ 1 , ..., ρ n +3 to b e equal to 0 , 1 , ∞ and the other n functions to b e equal to u 1 , ..., u n (cf. [5], Section 3). It w ould b e interesting to study the degenerate cases when some of the functions ρ i coincide (cf. [7], Section 5). The most symmetric case is giv en b y s 1 = ... = s n − k +1 = s n +1 = s n +2 = − k + 1 n + 3 , s n − k +2 = ... = s n = n − k + 2 n + 3 . 18 In this case all num b ers (4.42) are equal to − n − k +2 n +3 . P ossibly for n = 3 , k = 1 t hese v alues of parameters corresp o nd t o pseudop otentials for inte grable Lag rangians of the form L ( u x , u y , u z ) [20, 10 ] whereas fo r n = 5 , k = 3 they are relat ed to the integrable Hirota t yp e equations [13]. Example 5. Let n = 5, k = 3 and s 1 = s 2 = s 3 = s 6 = s 7 = − 1 2 , s 4 = s 5 = 1 2 . It turns out that there exists a basis g 1 , g 2 , g 3 , h 1 , h 2 , h 3 in H suc h that ∆( g 1 , h 1 , h 2 ) = ∆( g 3 , h 2 , h 3 ) , ∆( g 2 , h 2 , h 3 ) = ∆( g 1 , h 3 , h 1 ) , (4.43) ∆( g 3 , h 3 , h 1 ) = ∆( g 2 , h 1 , h 2 ) . Indeed, the syste m (4.43) is a conseque nce of equations g 1 h 1 ,u 4 − h 1 g 1 ,u 4 + g 2 h 2 ,u 4 − h 2 g 2 ,u 4 + g 3 h 3 ,u 4 − h 3 g 3 ,u 4 = 0 , g 1 h 1 ,u 5 − h 1 g 1 ,u 5 + g 2 h 2 ,u 5 − h 2 g 2 ,u 5 + g 3 h 3 ,u 5 − h 3 g 3 ,u 5 = 0 , (4.44) g 1 ,u 4 h 1 ,u 5 − h 1 ,u 4 g 1 ,u 5 + g 2 ,u 4 h 2 ,u 5 − h 2 ,u 4 g 2 ,u 5 + g 3 ,u 4 h 3 ,u 5 − h 3 ,u 4 g 3 ,u 5 = 0 . Consider t he system consisting of equations (4.44) and all its first and second deriv atives with resp ect to u 1 , ..., u 5 . Note that differen tiating (4.44), w e eliminate second deriv ativ es of h i and g i b y (1.8) , (1.9). One can c hec k that this system is inv ariant with resp ect to the deriv ations by u 1 , ..., u 5 . A t a fixed generic p oin t u 0 1 , ..., u 0 5 the system can b e regaded as a n algebraic v ariety for the v alues of g i , h i and their first deriv ativ es. It can b e c heck ed that this v ariet y consists of sev eral comp onen ts and the maximal dimension of the comp onent equals 24. Since the v ector fields ∂ ∂ u i are tangent to this v ariet y , an y its p oin t considered as the initial data defines the solutions g i , h i of (1.8), (1.9) suc h that the corresponding p oin t b elong s to the v ariet y for an y v alues of u 1 , ..., u 5 . It is p ossible to c heck that there exists an alg ebraic comp onen t o f dimension 21 of t he v ariety suc h t hat the W ronskian of g i , h i at u 0 1 , ..., u 0 5 is non-zero. Prop osition 9 and equations (4.43) allo w us to define a function z suc h that z t 1 ,t 1 = ∆( g 1 , h 2 , h 3 ) ∆( h 1 , h 2 , h 3 ) , z t 2 ,t 2 = ∆( g 2 , h 3 , h 1 ) ∆( h 1 , h 2 , h 3 ) , z t 3 ,t 3 = ∆( g 3 , h 1 , h 2 ) ∆( h 1 , h 2 , h 3 ) , z t 1 ,t 2 = ∆( g 2 , h 2 , h 3 ) ∆( h 1 , h 2 , h 3 ) = ∆( g 1 , h 3 , h 1 ) ∆( h 1 , h 2 , h 3 ) , z t 2 ,t 3 = ∆( g 3 , h 3 , h 1 ) ∆( h 1 , h 2 , h 3 ) = ∆( g 2 , h 1 , h 2 ) ∆( h 1 , h 2 , h 3 ) , z t 3 ,t 1 = ∆( g 1 , h 1 , h 2 ) ∆( h 1 , h 2 , h 3 ) = ∆( g 2 , h 1 , h 2 ) ∆( h 1 , h 2 , h 3 ) . It terms of t his f unction w e can rewrite the system (4.30), (4.31 ) , (4.32) a s a single equation of the form (1.11). Integrable system s of this form w ere studied in [1 3]. The pseudopo t entials considered ab o v e corresp ond t o the generic inte grable system of this form. Remark 6. It is easy t o see tha t the gro up S P 6 acts on the set of bases in H satisfying (4.44). This agrees with t he result of [13] that this group acts on the set of in tegrable equations of the form (1.1 1).  19 5 In tegr abl e (1+1) - dimension al systems of hydro dynamic t yp e In this section we consider inte grable (1+1)-dimensional h ydro dynamic t yp e systems (1.12) constructed in terms of g eneralized hypergeometric functions. T hese systems app ear as the so-called h ydro dynamic r eductions of pseudopotentials A n,k (see the next Section). By in- tegrabilit y w e mean the existence of infinite n umber of h ydro dynamic comm uting flow s and conserv a tion la ws. It is kno wn [21] that this is equiv a len t to the follo wing relations for the v elo cities v i ( r 1 , ..., r N ): ∂ j ∂ i v k v i − v k = ∂ i ∂ j v k v j − v k , i 6 = j 6 = k , (5.45) Here ∂ α = ∂ ∂ r i , α = 1 , . . . , N . The system (1.12) is called sem i-Hamiltonian if conditions (5.45) hold. The main geometrical ob ject related to a semi-Hamiltonia n system (1.12) is a diagonal metric g k k , k = 1 , . . . , N , where 1 2 ∂ i log g k k = ∂ i v k v i − v k , i 6 = k . (5.46) In view of (5.45 ) , the ov erdetermined system (5.46) is compatible and the function g k k is defined up to arbitrary f actor η k ( r k ). The metric g k k is called the metric asso ciate d to (1.12). It is kno wn that tw o hydrodynamic t yp e systems are compatible iff they p ossess a common asso ciated metric [21]. A diagonal metric g k k is called a metric of Egor ov typ e if for an y i, j ∂ i g j j = ∂ j g ii . (5.47) Note that if a Ego ro v- type metric asso ciat ed with a h ydro dynamic-type system of the form (1.12) exists, then it is unique. F or any Egorov’s metric there exists a p o ten tial G suc h that g ii = ∂ i G . Semi-Hamiltonian systems p ossessing asso ciated metrics o f Egorov type play imp orta nt role in the theory of WDVV asso ciativit y equations and in the theory of F rob enious manifolds [3, 16, 22 ]. Let w ( r 1 , ..., r N ) , ξ 1 ( r 1 , ..., r N ) , ..., ξ N ( r 1 , ..., r N ) b e a solution of (1.13). It can b e easily v erified that this system is in in v olution and therefore its solution admits a lo cal para meteri- zation by 2 N functions of one v ariable. L et u 1 ( r 1 , ..., r N ) , ..., u n ( r 1 , ..., r N ) be a set of solutions of the system (1.14). It is easy to ve rify that this system is in in v olution and therefore has an one-parameter family of solutions for fixed ξ i , w . Consider the follo wing system r i t = S n,k ( g 1 , ξ i ) S n,k ( g 2 , ξ i ) r i x , (5.48) 20 where g 1 , g 2 are linearly indep enden t solutions of (1.8), (1.9), the p olynomials S n,k , k > 0 are defined b y (4.25), and S n, 0 = S n (see form ula ( 3 .17)). Theorem 3. The system (5 .4 8) is semi-Hamiltonia n. The associat ed metric is give n b y g ii = S n ( g 2 , ξ i ) 2 ( ξ i − u 1 ) − 2 s 1 − 2 · · · ( ξ i − u n ) − 2 s n − 2 ξ − 2 s n +1 − 1 i ( ξ i − 1 ) − 2 s n +2 − 1 ∂ i w for k = 0 , and b y g ii = S n,k ( g 2 , ξ i ) 2 ( ξ i − u 1 ) − 2 s 1 − 2 · · · ( ξ i − u n − k +1 ) − 2 s n − k +1 − 2 × ( ξ i − u n − k +2 ) − 2 s n − k +2 · · · ( ξ i − u n ) − 2 s n ξ − 2 s n +1 − 1 i ( ξ i − 1) − 2 s n +2 − 1 ∂ i w for k > 0. Pro of. Subs tituting the expression for the metric in to (5.46), where v i are sp ecified by (5.48), one o btains the iden tit y by virtue of (1.13), (1.14).  Remark 7. The system (5.4 8) do es not p o ssess the asso ciated metric of the Egorov type in general. How ev er, for v ery sp ecial v alues of the parameters s i in (1.8), (1.9) there exists g 2 ∈ H suc h that the metric is of the Egoro v ty p e for all solutions of the system (1 .13), (1.14). F or instance, if the defect k equals zero, then this happ e ns exactly in the f o llo wing cases: s i = 0 for all i ; s l = − 1 fo r some l and s i = 0 for i 6 = l ; s l = − 1 2 for some l and s i = 0 for i 6 = l ; s j = s l = − 1 2 for some j 6 = l and s i = 0 for i 6 = j, i 6 = l . Prop osition 10. Supp ose that a solution ξ i , w of (1.13) and solutions u 1 , ..., u n of (1.14) are fixed. Then the h ydro dynamic type systems r i t 1 = S n,k ( g 1 , ξ i ) S n,k ( g 3 , ξ i ) r i x , r i t 2 = S n,k ( g 2 , ξ i ) S n,k ( g 3 , ξ i ) r i x (5.49) are compatible for all g 1 , g 2 . Pro of. Indeed, the metric asso ciated with (5.48) do es not depend on g 2 . Therefore the systems (5.49) ha s a common metric dep ending on g 3 and on solutions of (1.13), (1.14).  Remark 8. One can also construct some compatible systems of the form (5.49) using Prop osition 5. Set g 2 = Z ( u 1 , ..., u n , u n +1 ) in (5 .4 9). Here u n +1 is an arbitra ry solution of (1.14) distinct from u 1 , ..., u n . It is clear that the flo ws (5.4 9) are compatible for suc h g 2 and an y g 1 ∈ H . Moreo v er, Propo sition 5 implies that the flo ws (5.49) are compatible if w e set g 1 = Z ( u 1 , ..., u n , u n +1 ), g 2 = Z ( u 1 , ..., u n , u n +2 ) for t wo arbitrary solutions u n +1 , u n +2 of (1.14). All mem b ers of the hierarch y constructed in Pro p osition 10 p ossess a disp ersionless Lax represen tation (1.2) with common L ( p, r 1 , ..., r N ). Define a function L ( ξ , r 1 , ..., r N ) by the follo wing system ∂ i L = ξ ( ξ − 1) ∂ i w L ξ ξ − ξ i , i = 1 , ..., N . (5.50) 21 Note that the system (5 .5 0) is in in v olution and therefore the f unction L is defined uniquely up to inessen tial transforma t io ns L → λ ( L ). T o find the function L ( p, r 1 , ..., r N ) one ha s to express ξ in terms of p b y (3.21) for k = 0 or b y ( 4.29) for k > 0. Prop osition 11. Let u 1 , . . . , u n b e arbitr ary solution of (1.14) . Then system (5.48) admits the disp ersionless Lax represen tat ion (1.2), where A = A n,k is defined b y (3 .21) fo r k = 0 and b y (4.29) for k > 0. Pro of. Substituting A = A n,k defined b y (3.21 ) for k = 0 and b y (4.29) fo r k > 0 into (1.2) and calculating L t b y virtue of (5.48) w e arriv e to the expression ∂ i L = ∂ i P n,k ( g 2 , ξ ) · S n,k ( g 1 , ξ i ) − ∂ i P n,k ( g 1 , ξ ) · S n,k ( g 2 , ξ i ) P n,k ( g 2 , ξ ) ξ · S n,k ( g 1 , ξ i ) − P n,k ( g 1 , ξ ) ξ · S n,k ( g 2 , ξ i ) L ξ . T aking into accoun t the equation P n,k ( g i , ξ ) ξ = S n,k ( g i , ξ )( ξ − u 1 ) − s 1 − 1 ... ( ξ − u n − k +1 ) − s n − k +1 − 1 ( ξ − u n − k +2 ) − s n − k +2 ... ( ξ − u n ) − s n ξ − s n +1 − 1 ( ξ − 1) − s n +2 − 1 and writing down P n,k ( g i , ξ ) u m in terms of P n,k ( g 1 , ξ ) u n − k +1 , ..., P n,k ( g 1 , ξ ) u n b y (4.27) , (4.28), one can readily c hec k this equation.  Let the function ξ ( L, r 1 , ..., r N ) b e in vers e to L ( ξ , r 1 , ..., r N ) . It is easy to c hec k that u = ξ ( L, r 1 , ..., r N ) , where L plays a role of arbitrary parameter, satisfies (1.14). As usual, the Lax represen ta tion defines conserv ed densities, common for the whole hierar- c hy , b y formula (1.3 ) . Since our definition of A n,k is parametric, w e can refo r m ulate this fact as Prop osition 12. Supp ose (5.48) is defined by solutions u 1 , . . . , u n of the system (1.14). Let U be an y solution of (1.14). Then ∂ ∂ t P n,k  g 2 ( u 1 , u 1 , . . . , u n ) , U  = ∂ ∂ x P n,k  g 1 ( u 1 , u 1 , . . . , u n ) , U  is a conserv ation law for (5.48). Since the generic solution U dep ends on a parameter, we hav e constructed an one-parametric family of common conserv ation law s for our hierarc hy (5.48) of hydrodynamic type systems. 6 Hydro dynamic r e ductio n s and in tegrability In this section w e sho w that in tegr a ble (1+1)-dimensional systems constructed in Section 5 define h ydro dy namic reductions for pseudop otentials and 3- dimensional systems from Sections 3 and 4. F ollow ing [17, 15, 7], we giv e a definition of integrabilit y fo r equations (1.2), (1.4) and (1.5) in t erms o f h ydro dynamic reductions. Supp ose there exists a pair of compatible semi-Hamiltonian h ydro dy namic-t yp e systems of the form r i t 1 = v i 1 ( r 1 , ..., r N ) r i x , r i t 2 = v i 2 ( r 1 , ..., r N ) r i x (6.51) 22 and functions u i = u i ( r 1 , ..., r N ) suc h that these functions satisfy (1.5) for any solution of (6.51). Then ( 6.51) is called a hydr o dynamic r e duction for (1.5). Definition 1 [17 ]. A system of the form (1.5) is called inte gr able if equation (1 .2) p oss esses sufficien tly many h ydro dynamic r eductions fo r each N ∈ N . ”Sufficien tly man y” means that the set of h ydro dynamic reductions can b e lo cally parameterized b y 2 N functions of one v ariable. Note that due to gauge tr a nsformations r i → λ i ( r i ) w e hav e N essen tial functional parameters for h ydro dynamic reductions. Supp ose there exists a semi-Hamiltonian h ydro dynamic -t yp e system (1.12) and functions u i = u i ( r 1 , ..., r N ), L = L ( p, r 1 , ..., r N ) suc h that these functions satisfy disp ersionless Lax equation (1.2) for an y solution r 1 ( x, t ) , ..., r N ( x, t ) of the system (1.12). Then (1.12) is called a hydr o dynamic r e duction for (1.2). Definition 2 [7]. A disp ersionless Lax equation (1.2) is called inte gr able if equation (1.2) p ossesses sufficien tly many h ydro dynamic reductions for eac h N ∈ N . W e also call the corresp onding pseudop o ten tial A ( p, u 1 , ..., u n ) in tegrable. Example 6. Let us sho w that A = ln( p − u ) is integrable. Let w ( r 1 , ..., r N ), p i ( r 1 , ..., r N ), i = 1 , ..., N b e an arbitrary solution of the f ollo wing system (the so-called Gibb ons-Tsarev system [14]) ∂ j ξ i = ∂ j w ξ j − ξ i , ∂ ij w = 2 ∂ i w ∂ j w ( ξ i − ξ j ) 2 , i, j = 1 , ..., N , i 6 = j. (6.52) It is easy to v erify that this system is in in volution and therefore its general solution admits a lo cal parameterizations by 2 N functions o f o ne v ariable. D efine a function L ( p, r 1 , ..., r N ) by the follo wing system ∂ i L = ∂ i w L p p − ξ i , i = 1 , ..., N . (6.53) This system is in in volution and therefore defines the function L uniquely up to inessen tial transformations L → λ ( L ). Finally , let u ( r 1 , ..., r N ) b e a solution of t he system ∂ i u = ∂ i w ξ i − u , i = 1 , ..., N . (6.54) It is easy to c heck that the system ( 6.54) is in inv olution. It f ollo ws from (6.5 2 ), (6.53), (6.54) that the sys tem r i t = 1 ξ i − u r i x (6.55) is a hydrodynamic reduction of equation ( 1 .2) with A = ln( p − u ). Remark 9. The standard fo rm for the Gibb ons-Tsarev system [1 5] related to h ydro dy namic reductions is giv en b y ∂ i ξ j = F ( ξ i , ξ j , u 1 , . . . , u n ) ∂ i u n , ∂ i ∂ j u n = H ( ξ i , ξ j , u 1 , . . . , u n ) ∂ i u n ∂ j u n , i 6 = j 23 ∂ i u l = G l ( ξ i , u 1 , . . . , u n ) ∂ i u n , l < n. Here i, j = 1 , ..., N , u l ( r 1 , . . . , r N ) are the functions, whic h define the reduction, and ξ i ( r 1 , . . . , r N ) are some a uxiliary functions. T o bring (6.52), (6.54) to this form, one has to eliminate the additional unkno wn w . The result is giv en by ∂ j ξ i = ξ i − u ξ j − ξ i ∂ j u, ∂ i ∂ j u = ξ i + ξ j − 2 u ( ξ i − ξ j ) 2 ∂ i u∂ j u. (6.56) In this case n = 1 , u 1 = u. There is t he following generalization of (6.56 ) to t he case of arbitra ry p olynomial P ( x ) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 and arbitrary n : u 1 − ξ i P ( u 1 ) ∂ i u 1 = ... = u n − ξ i P ( u n ) ∂ i u n , i = 1 , ..., N , ∂ ij u n = K 2 ( ξ i , ξ j ) u 2 n + K 1 ( ξ i , ξ j ) u n + K 0 ( ξ i , ξ j ) P ( u n )( ξ i − ξ j ) 2 ∂ i u n ∂ j u n , ∂ i ξ j = P ( ξ j )( u n − ξ i ) P ( u n )( ξ i − ξ j ) ∂ i u n , i, j = 1 , ..., N , i 6 = j, (6.57) where K 2 ( ξ i , ξ j ) = 2 a 3 ( ξ i − ξ j ) 2 , K 1 ( ξ i , ξ j ) = − a 3 ( ξ 2 i ξ j + ξ i ξ 2 j ) + a 2 ( ξ 2 i + ξ 2 j − 4 ξ i ξ j ) − a 1 ( ξ i + ξ j ) − 2 a 0 , K 0 ( ξ i , ξ j ) = 2 a 3 ξ 2 i ξ 2 j + a 2 ( ξ 2 i ξ j + ξ i ξ 2 j ) + a 1 ( ξ 2 i + ξ 2 j ) + a 0 ( ξ i + ξ j ) . Using transfor ma t io ns of the form u i → au i + b cu i + d , ξ i → aξ i + b cξ i + d , o ne can put the p olynomial P to one o f the canonical forms: P ( x ) = x ( x − 1), P ( x ) = x , or P ( x ) = 1. If P ( x ) = 1 , then (6.57) with n = 1 coincides with ( 6 .56). F orm ula s (1 .13), (1.14) are equiv alen t to (6.57), where P ( x ) = x ( x − 1). Definition 3. Tw o in tegra ble pseudop oten t ia ls A 1 , A 2 are called c omp atible if the system L t 1 = { L, A 1 } , L t 2 = { L, A 2 } p ossesses sufficien tly many h ydro dynamic reductions (6.5 1 ) for each N ∈ N . If A 1 , A 2 are compatible, then A = c 1 A 1 + c 2 A 2 is integrable for any constan ts c 1 , c 2 . Indeed, the syste m r i t = ( c 1 v i 1 ( r ) + c 2 v i 2 ( r )) r i x is a hydrodynamic reduction of (1.2). Example 7. The functions A 1 = ln( p − u 1 ) and A 2 = ln( p − u 2 ) are compatible. Moreov er, A = c 1 ln( p − u 1 ) + ... + c n ln( p − u n ) is integrable for an y constants c 1 , ..., c n . Indeed, let w , p i satisfy (6.52) and u 1 , u 2 b e tw o differen t solutions of (6.54). It is easy to c hec k that the corresp onding flo ws are compatible b y virtue of (6.52), (6.53), (6.54). 24 Definition 4. By 3 - dimensional system asso ciat ed with compatible functions A 1 , A 2 w e mean the sys tem of the form (1.5) equiv alent to compatibilit y conditions fo r the system ψ t 2 = A 1 ( ψ t 1 , u 1 , ..., u n ) , ψ t 3 = A 2 ( ψ t 1 , u 1 , ..., u n ) . (6.58) It is clear that any system asso ciat ed with a pair of compatible functions p ossesses suffi- cien tly man y h ydro dynamic reductions and therefore it is in tegrable in the sense of Definition 1. Example 8. Let A 1 = ln( p − u ) and A 2 = ln( p − v ). The asso ciated 3-dimensional system has the form u t 3 = v t 2 , v t 1 − v u t 3 = u t 1 − uv t 2 . The follo wing statemen t is a reform ulat ion of Prop osition 11. Theorem 4. The system (5.48) is a hydrodynamic reduction of the pseudop otential A n,k defined by (3.21) if k = 0 and by (4.29) if k > 0. Recall that w e use the nota tion S n ≡ S n, 0 , A n ≡ A n, 0 , P n ≡ P n, 0 . Prop osition 13. Supp ose g 1 , g 2 , g 3 , h 1 , ..., h k ∈ H are linearly indep enden t. Define pseu- dop oten tials A 1 , A 2 b y A 1 = P n,k ( g 1 , ξ ) , A 2 = P n,k ( g 2 , ξ ) , p = P n,k ( g 3 , ξ ) . Then A 1 and A 2 are compatible. Pro of. Note that the system (1.13), (5 .50) do es not dep end on g 1 , g 2 , g 3 and t herefore w e ha ve a family of functions L, ξ i , u i whic h giv e h ydro dyn amic reduction of the for m (5.48) for b oth A 1 and A 2 . Moreo v er, a ccording to Prop o sition 10 the flo ws r i t 1 = S n,k ( g 1 , ξ i ) S n,k ( g 3 , ξ i ) r i x , r i t 2 = S n,k ( g 2 , ξ i ) S n,k ( g 3 , ξ i ) r i x are compatible.  Remark 10. This result implies that 3-dimensional h ydro dyn amic ty p e systems con- structed in Sections 4 , 5 po ssess sufficien t ly man y hy dro dynamic r eductions. Remark 11. Using prop osition 5, one can construct compatible pseudop otentials depend- ing o n differen t n um b er of u i . Indeed, let g 1 , g 3 , h 1 , ..., h k ∈ H and g 2 = Z ( u 1 , ..., u n , u n +1 ). Then A 2 dep ends on u 1 , ..., u n , u n +1 and A 1 dep ends on u 1 , ..., u n only . 7 Conclus ion All known in tegrable pseudop otentials A ( p, u 1 , ..., u n ) satisfy the pro p ert y P  A ppp A 2 pp , A p  = 0 , 25 where P ( x, y ) is a p olynomial in x, y with co efficien ts dep ending on u 1 , ..., u n . In this sense an y pseudop ot ential A is asso ciated with the a lg ebraic curve E = { ( x, y ) ∈ C 2 ; P ( x, y ) = 0 } . Moreo ve r, compatible pseudopo ten tials are asso ciated to isomorphic curv es. If a 3- dimensional disp ersionless sys tem is constructed b y t w o compatible pseudopo ten tials, then this curve is isomorphic to the so-called spectral curve (see [17]) of the sys tem. In this pap er we hav e constructed a wide class of in tegrable pseudop o ten tials associated with rational curv es. W e b eliev e that all pseudopo t entials asso ciated with rational curv es can b e obtained as a limit from our pseudop otentials. W e are going to describe all suc h limits in a separate pap er. It is kno wn [2] that pseudop otentials asso ciated with curv es of higher genus a lso exist. It is lik ely that one can describ e all pseudop otentials asso ciated with the elliptic curv e in a similar manner to the w ay w e hav e done the rationa l case in this pap er. W e ar e go ing to consider this problem in the next pap er. Ac kno wledgmen t s. Authors thank B.A. D ubro vin, E.V. F erap onto v, I.M. Kric hev er, O.I. Mokho v, and M.V. P avlo v for fruitf ul discussions. V.S. thanks IHES a nd A.O. thanks MPIM and IHES for hospitality and financial supp ort. V.S. w as partially supp orted by the RFBR gran ts 08-01-46 1 and NS 3472 .2008.2. References [1] V.E. Zakhar ov, A.B. Shab at , In tegration of non-linear equations of mathematical phy sics b y t he inv erse scattering metho d, F unc. Anal. and Appl. 13 (3) (1979) 1 3–22. [2] I.M. Krichever , The τ -function of the univ ersal Whitham hierarch y , matrix mo dels and top ological field theories, Comm. Pure Appl. Math., 47 (1994), no. 4, 4 37–475. [3] B.A. Dubr ovi n , Geometry of 2D to p ological field theories. I n Inte gr able Systems and Quantum Gr oups , Lecture Notes in Math. 1620 (1996), 120–348. [4] A. Odess kii, V. Sokolov , On (2+1)- dimensional hy dro dynamic-ty p e sy stems p ossessing pseudop o ten tial with mo v able singularities, to app e ar in F unc. A n al. Appl. [5] A.V. Odesskii , A family of (2+1)-dimensional hy dro dynamic-ty p e systems p ossessing pseudop o ten tial, arXiv:0704.357 7v3 [math. AP], to app ear in Selecta Mathematica. [6] M.V. Pavlov , Classification of the Egoro v hy dro dynamic chains. Theor. Math. Phy s. 138 No. 1 (2004) 55 -71. [7] A. Odes skii, M.V. Pavlov and V.V. Sok olov , Classification of in tegra ble Vlaso v-type equa- tions, arXiv:0710.5655, Theor. Math. Ph ys. 154 (2)(2 008) 209-219 . [8] H. Bateman, A. Er delyi , Higher transcenden tal functions, 195 3 . 26 [9] I.M. Gelfand, M.I. Gr ae v, V.S. R etakh , General h yp ergeometric systems of equations and series of hypergeometric t yp e, Russian Math. Surv eys 47 (1992), no. 4, 1–88 [10] E.V. F er ap on tov, A.V. Odesskii , In tegrable Lag r a ngians and mo dular forms, [11] P.A. Bur ovskiy, E.V. F er ap on tov and S .P. Tsar ev , Second or der quasilinear PDEs and conformal structures in pro jectiv e space, arXiv:0802.26 2 6 [nlin. SI]. [12] W. F. Ames, R. L. Anderson, V. A. Dor o dnitsyn, E. V. F er ap ontov, R. K. Gaziz ov, N. H. Ibr agimov, S. R. Svirshchevskii , CR C handb o ok of L ie group analysis of differen tial equations. V ol. 1. Symmetries, exact solutions and conserv at io n law s. CR C Press, Bo ca Raton, FL, 1994. [13] E.V. F er a p ontov, L. Hadji kos, K.R. K husnutdinova , In t egr a ble equations o f the disp er- sionless Hirot a t yp e and h yp ersurfaces in the Lag r a ngian G rassmannian, a r Xiv:07 0 5.1774. [14] J. Gibb ons, S .P. T sar ev , Reductions of Benney’s equations, Ph ys. Lett. A, 211 (1996) 19-24. J. Gibb ons, S.P. Ts ar ev , Conformal maps and reductions o f the Benney equations, Ph ys. Lett. A, 258 (19 99) 2 6 3-270. [15] M.V. Pavlov , Algebro-geometric approach in the theory of in t egrable h ydro dynamic-t yp e systems . Comm. Math. Ph ys., 272 (2) (2007) 469- 505. [16] A. A. Akhmetshin, I. M. Kric hever, Y. S. V olvovski , A g enerating form ula for solutions of asso ciativit y equations. Russian Math. Surv eys 54 (1 999), no. 2, 427–429. [17] E.V. F er ap ontov, K.R. Khusnutdinova , On in tegrability of (2+1)-dimensional quasilinear systems , Comm. Math. Ph ys. 248 ( 2 004) 187-206 , E.V. F er ap ontov, K.R. Khusnutdinova , The c haracterization of 2- comp onen t (2 + 1)-dimensional in tegra ble sys tems of hy dro dy- namic t yp e, J. Ph ys. A: Math. Gen. 37 (8) (2004) 2949–2963. [18] E.V. F er ap ontov, D.G. Marshal , Differen tial- g eometric approac h to the in t egr a bilit y of h ydro dy namic chains: the Haanties t ensor, Math. Ann. 339 (1) , ( 2007) 61–99. [19] M.V. Pavlov , Classification of in tegra ble h ydro dynamic c hains a nd generating functions of conserv atio n la ws, J. Phys . A: Math. Gen. 39 (34) (2006) 10803–108 1 9. [20] E.V. F er a p ontov, K.R. Khusnutdinova and S.P. Tsar ev , On a class of three-dimensional in tegrable Lagrangians, Comm. Math. Ph ys. 261 , no. 1 (2006) 225–243. [21] S .P. T sar ev , On P oisson bra c k ets and one-dimensional Hamiltonian systems of hydro- dynamic type, Soviet Math. Dokl., 31 (1985) 4 88–491. S.P. Tsar e v , The geometry of Hamiltonian systems of h ydro dy namic t yp e. The generalized ho dogra ph metho d, Math. USSR Izv estiy a, 37 No. 2 (1991) 397–419. 1048–1068. 27 [22] M.V. Pavlov, S.P. Tsar ev , T ri-Hamiltonian structures of the Egoro v systems of h ydro dy - namic t yp e. F unc. Anal. and Appl. , 37 ( 1 ) (2003) 32-45. 28