Integrability conditions for two-dimensional lattices

In tegrability conditi o ns for t w o-dimensi onal latti c e s I T Habibullin 1 , 2 , M N Kuznetso v a 1 and A U Sakiev a 3 1 Institute of Mathematics, Ufa F eder al Research Cent re , Russia n Aca dem y o f Sciences, 112 Chernyshevsky Street, Ufa 4 50008 , Russian F ederation 2 Bashkir State Universit y , 32 V a lidy Str e e t, Ufa 4500 76 , Russian F ederatio n 3 State Budgetary Educationa l Institution, G. Alm ukhametov Re publica n Bo arding School, 25/1 Z orge Street, Ufa 450 059, Russia n F eder ation E-mail: habibu llini smagil@gmail.com , mariya .n.ku znetsova@gmail.com and alfya. sakie va@yandex.ru Abstract. In the article some algebraic prop erties of nonlinear tw o-dimensio nal lattices of the form u n,xy = f ( u n +1 , u n , u n − 1 ) are s tudied. The problem of e x haustive description of th e in tegra ble cases o f this kind la ttices remains open. By using the approa ch, dev elop ed and tested in our previo us works w e adopted the metho d of c haracteris tic Lie-Rineha rt algebras to this case. In the article w e deriv ed a n effective integrability conditions for the lattice and proved that in the integrable ca se the function f ( u n +1 , u n , u n − 1 ) is a quasi-p olyno mial satisfying the following equation ∂ 2 ∂ u n +1 ∂ u n − 1 f ( u n +1 , u n , u n − 1 ) = C e αu n − αm 2 u n +1 − αk 2 u n − 1 , where C a nd α are constant parameters and k , m are nonnega tive integers. P ACS num be r s: 02.30.Ik Inte gr a b ility c onditions for two-dim ensional lattic es 2 1. In tr o duction Multidimensional integrable equations, suc h as the KP and D a v ey-Stew artson equations, the tw o-dimensional T o da lattice and so on hav e imp orta n t applicatio ns in ph ysics and geometry . In recen t y ears, v arious metho ds hav e b een dev elop ed to study suc h kind equations (see, f o r instance, [1]-[5]). In this pap er w e consider a class of the lattices of the form u n,xy = f ( u n +1 , u n , u n − 1 ) , −∞ < n < ∞ , (1.1) where t he unkno wn u n = u n ( x, y ) dep ends on the real x, y and the integer n . The function f o f three v ariables is assumed to b e analytic in a domain D ⊂ C 3 . Recall that class (1.1) con tains suc h a famous equation as the tw o-dimensional T oda lattice, whic h can b e represen ted as (1 .1) in the following three w a ys u n,xy = e u n +1 − 2 u n + u n − 1 , (1.2) v n,xy = e v n +1 − v n − e v n − v n − 1 , (1.3) w n,xy = e w n +1 − 2 e w n + e w n − 1 . (1.4) The lattices are related to eac h other b y the linear substitutions w n = v n +1 − v n , v n = u n − u n − 1 . An imp or tan t op en problem is to describ e all of the in tegrable cases in t he class (1.1). The purp ose of the presen t article is to give an appropriate algebraic formalization of the classification problem and deriv e effectiv e necessary integrabilit y conditions for (1.1). Our inv estigation is based on the classification sc heme, outlined in [6]-[10]. The sc heme realizes the commonly accepted opinion, that an y integrable equation in 3D admits a large set of in tegrable in a sense, tw o dimensional r eductions. The h ydro dynamic r eduction metho d, dev elop ed in [11]-[15] is one of the fruitful applications of this idea. According to t he metho d 3D equation is in tegrable if it admits sufficien tly large class of 2D reductions in the fo rm of the in tegrable h ydro dynamic ty p e systems. In o ur study w e observ ed that existence of a sequence o f Da rb oux in tegrable reductions migh t b e regarded as a sign of integrabilit y as well. In the case of lattice (1.1) we use the follo wing Definition 1. L attic e (1.1 ) is c al le d inte gr able if ther e exist lo c al ly an alytic functions ϕ and ψ of t wo variables such that for any choic e of the inte gers N 1 , N 2 the hyp erb olic typ e system u N 1 ,xy = ϕ ( u N 1 +1 , u N 1 ) , u n,xy = f ( u n +1 , u n , u n − 1 ) , N 1 < n < N 2 , (1.5) u N 2 ,xy = ψ ( u N 2 , u N 2 − 1 ) . obtaine d fr om lattic e (1.1 ) is inte gr able in the sense of Darb oux. The abov e integrable mo dels (1.2)-(1.4) are certainly in tegrable in the sense of our Definition 1 as w ell. Recall that Darbo ux in tegrabilit y means that the system admits a complete set o f nontrivial integrals in b oth c haracteristic directions of x and Inte gr a b ility c onditions for two-dim ensional lattic es 3 y . An effectiv e criterion o f suc h kind of in tegrabilit y is form ulated in terms of the c haracteristic Lie-Rinehart alg ebras (see Theorem 1 b elow in § 3). Prop erties of the c haracteristic algebra and its applicatio n fo r 1+1 dimensional con tin uous and discrete Darb oux inte gra ble systems a r e studied in [1 6]-[24]. In the contex t of 2+1 dimensional mo dels these algebras for the first time ha v e b een considered in [25]. In our recen t w orks [6]-[10 ] w e studied t he classification pro blem for t w o-dimensional lattices of a sp ecial kind, where the metho d of D arb oux in tegrable reductions and the algebraic approac h were used as basic implemen ts. W e note that discussion on the alternative approac hes to the study of Darb oux in tegrable equations based on the higher symmetries, the Laplace in v arian ts, etc. can b e found in [26 ]-[29]. The a rticle is organized as follows . In § 2 w e recall necessary definitions, intro duce the notion of the c haracteristic Lie-Rinehart algebra and form ulate an algebraic criterion of the Darb oux integrabilit y . In the third section w e study the connection b et w een the structure of the c haracteristic algebra and the prop erties of the func tion f ( u n +1 , u n , u n − 1 ) assuming that the reduced system (1 .5) is in tegrable in the sense o f Darb o ux. W e pro v e that f is a quasi-p olynomial with resp ect to all three arguments a nd give the complete description of its second order deriv at iv e f u n +1 ,u n − 1 . In the last fourth section w e deriv e the main result of the article b y formulating the necessary conditio n of in tegrability for the lattice (1.1). Some tediously long pro ofs carried ov er to the App endix . 2. Preliminaries No w let us recall some basic notions of the in tegrabilit y theory . A f unction I = I ( x, ¯ u, ¯ u x , ¯ u xx , ... ) dep ending on a finite n um b er of the dynamical v ariables is called an y - in tegral of system (1.5) if it solv es equation D y I = 0. Here ¯ u is a vec tor ¯ u = ( u N 1 , u N 1 +1 , . . . , u N 2 ), ¯ u x is its deriv ativ e and so on. Similarly , a function J = J ( y , ¯ u, ¯ u y , ¯ u y y , ... ) is a x -inte gra l if the equation D x J = 0 holds. Integrals of the form I = I ( x ) and J = J ( y ), dep ending o nly o n x and y are called trivial. A system (1.5) is called in tegrable in the sense of D a rb oux if it admits a complete set of functionally indep enden t no ntrivial in tegrals in b oth c haracteristic directions x and y . Actually it is required that the nu mber of functionally indep enden t in tegrals is N 1 + N 2 − 1 in eac h direction. Let us tak e a non trivial y -in tegral I = I ( ¯ u, ¯ u x , ¯ u xx , ... ) of the s ystem (1 .5). Obviously the op erator D y acts on I a s a v ector field of the following fo rm: D y = N 2 X j = N 1  u j,y ∂ ∂ u j + f j ∂ ∂ u j,x + D x ( f j ) ∂ ∂ u j,xx + · · ·  , (2.1) where f j = f ( u j +1 , u j , u j − 1 ). The v ector field (2.1) in a natural w ay splits down in to a linear com bination of the indep endent op erators X j and Z D y = N 2 X j = N 1 u j,y X j + Z, Inte gr a b ility c onditions for two-dim ensional lattic es 4 where X j = ∂ ∂ u j , Z = N 2 X j = N 1  f j ∂ ∂ u j,x + D x ( f j ) ∂ ∂ u j,xx + · · ·  . (2.2) Ob viously equation D y I = 0 righ t aw a y implies that X j I = 0 and Z I = 0. W e study D arb oux integrable systems b y using characteristic algebras. Denote by A the ring o f lo cally analytic functions o f the dynamical v ariables ¯ u, ¯ u x , ¯ u xx , . . . . Let us introduce the Lie algebra L y with the usual op eration [ W 1 , W 2 ] = W 1 W 2 − W 2 W 1 , generated b y the differen tial op erators Z and X j defined in (2.2) ov er the ring A . W e assume the consistency conditions: 1). [ W 1 , aW 2 ] = W 1 ( a ) W 2 + a [ W 1 , W 2 ], 2). ( aW 1 ) b = aW 1 ( b ) to b e v alid for an y W 1 , W 2 ∈ L y and a, b ∈ A . In other w ords w e request tha t, if W 1 ∈ L y and a ∈ A then aW 1 ∈ L y . In suc h a case the algebra L y is called the Lie- Rinehart algebra [30], [31]. W e call it also the characteristic algebra in the direction o f y . In a similar w ay the c haracteristic alg ebra L x in t he direction of x is defined. The alg ebra L y is of a finite dimension if it admits a basis consisting of a finite n um b er of t he elemen ts Z 1 , Z 2 , . . . , Z k ∈ L y suc h that an arbitrary op erator Z ∈ L y is represen ted as a linear com bination of the form Z = a 1 Z 1 + a 2 Z 2 + . . . a k Z k , where the co efficien ts are functions a 1 , a 2 , . . . , a k ∈ A . No w w e are ready to form ulate an algebraic criterion of the integrabilit y of a h yp erb olic ty p e system in the sense of Dar b oux [20, 21], whic h pla ys a crucial role in our in ve stigation. Theorem 1. System (1.5) admits a c omplete set of the y -inte gr als (a c omplete set of the x -inte gr als) if and on l y if its char acteristic algebr a L y (r esp e ctively, its char acteristic algebr a L x ) is of a finite dime nsion. Corollary of Theorem 1. Syst em (1.5) is inte gr able in the sen se of D arb oux if and only if b oth char acteristic algebr as L x and L y ar e of a finite dimension. 3. In vestigation of the c haracteristic algebras Assume that la ttice (1.1) is in tegrable in the sense of Definition 1. Then for any pair of the in tegers N 1 , N 2 h yp erb olic t yp e system (1.5) has to b e Darb oux in tegrable. Therefore according to Theorem 1 the algebras L x and L y m ust b e o f a finite dimension. Since the elemen ts of the algebras are v ector fields with an infinite num ber o f comp onen ts, the problem of clarifying the linear dep endence of a set of elemen ts b ecomes v ery non- trivial. T o this aim the follo wing lemma pro vides a ve ry useful implemen t [1 6], [20]. Lemma 1. If the ve ctor field of the form Z = N 2 X i = N 1 z 1 ,i ∂ ∂ u i,x + z 2 ,i ∂ ∂ u i,xx + · · · Inte gr a b ility c onditions for two-dim ensional lattic es 5 solves the e quation [ D x , Z ] = 0 , then Z = 0 . Lemma 2. The fol lowin g formulas h old: [ D x , X j ] = 0 , [ D x , Z ] = − N 2 X j = N 1 f j X j . Pro of of Lemma 2. The op erator D y acts on the function F ( ¯ u, ¯ u x , ¯ u xx , . . . ) b y the follow ing rule D y = Z + N 2 X j = N 1 u j,y X j . Since the op erators D x , D y comm ute with one another, w e ha v e the relation: [ D x , D y ] = [ D x , Z + N 2 X j = N 1 u j,y X j ] = 0 . Using prop erties of the comm utators, w e get [ D x , Z ] + N 2 X j = N 1 f j X j + N 2 X j = N 1 u j,y [ D x , X j ] = 0 . By comparing the co efficien ts in front of the indep enden t v ariables u j,y , w e easily o btain the statemen t of the Lemma 2. Let us construct a sequence of the op erators Z 0 , Z 1 , Z 2 , . . . by setting Z 0 = [ X 0 , Z ] , Z 1 = [ X 0 , Z 0 ] , . . . , Z j +1 = [ X 0 , Z j ] , . . . It is easy to c hec k that the relations ho ld [ D x , Z j ] = − 1 X k = − 1 X j +1 0 ( f k ) X k , (3.1) where f 0 = f ( u 1 , u 0 , u − 1 ). Lemma 3. Supp ose that lattic e (1.1) is inte gr able in the sense of Definiton 1, then the function f = f ( u 1 , u 0 , u − 1 ) is a quasi-p olynomial with r esp e ct to any of its ar guments u − 1 , u 0 , u 1 . Pro of. W e supp ose that lattice (1.1 ) is integrable in the sense of D efinition 1 , then the c haracteristic alg ebra should b e finite-dimensional. That is why there exists a natural M suc h that Z M +1 is linearly expressed through the previous mem b ers of the sequence : Z M +1 = M X i =0 λ i Z i , (3.2) where the o p erators Z 1 , Z 2 , . . . Z M are linearly indep enden t. W e comm ute b o th sides o f equalit y (3.2 ) with the op erato r D x and due to (3.1 ) we arriv e at: − 1 X j = − 1 X M +2 0 ( f j ) X j = M X i =0 ( D x ( λ i ) Z i − λ i 1 X k = − 1 X i +1 0 ( f k ) X k ) . Inte gr a b ility c onditions for two-dim ensional lattic es 6 Comparing the co efficien ts b efore linearly indep enden t op erato r s Z i for i = 0 , 1 , ..., M one gets D x ( λ i ) = 0 and therefore λ i = const . Comparison of the factors b efore X j yields for j = − 1 , 0 , 1: ( X M +1 0 − λ M X M − λ M − 1 X M − 1 − ... − λ 0 ) X 0 ( f j ) = 0 . Th us all three functions f − 1 = f ( u 0 , u − 1 , u − 2 ), f 0 = f ( u 1 , u 0 , u − 1 ) and f 1 = f ( u 2 , u 1 , u 0 ) are quasi-p olynomials on the v ariable u 0 , hence eviden tly f ( u 1 , u 0 , u − 1 ) is a quasi- p olynomial with r esp ect to all of its argumen ts. Lemma 3 is pr ov ed. Lemma 4. Op er ator W 0 = [ X 1 , [ X − 1 , Z ]] satisfies the c ondition [ D x , W 0 ] = − f u 1 u − 1 X 0 . Pro of. Lemma 4 is easily pr ov ed b y using the Jacobi iden tity and formula [ D x , [ X 1 , Z ]] = − 2 X k =0 X 1 ( f j ) X j . Let us now construct a sequenc e of the form: W 0 , W 1 = [ X 0 , W 0 ] , W 2 = [ X 0 , W 1 ] , . . . , W k +1 = [ X 0 , W k ] , ... (3.3) Elemen ts o f t he sequence satisfy the form ulas: [ D x , W k ] = − X k 0 ( f u 1 u − 1 ) X 0 . (3.4) Since the c haracteristic algebra is finite-dimensional there exists a na tural M suc h that W M +1 is linearly expressed through the previous members: W M +1 + λ M W M + · · · + λ 1 W 1 + λ 0 W 0 = 0 , where W 0 , . . . , W M are linearly indep enden t. W e comm ute b o t h sides of this equality with the o p erator D x and apply formula (3.4). Th us w e obtain the relations D x ( λ j ) = 0 satisfied for j = 0 , 1 , ..., M and an equation X M +1 0 ( f u 1 u − 1 ) X 0 + λ M X M 0 ( f u 1 u − 1 ) X 0 + · · · + λ 0 f u 1 u − 1 X 0 = 0 . Ob viously the la tter implies:  X M +1 0 + λ M X M 0 + · · · + λ 0  f u 1 u − 1 = 0 . Let us denote through Λ( λ ) the characteristic p olynomial of this linear ordinary differen tial equation, i.e. Λ( λ ) := λ M +1 + λ M λ M + · · · + λ 0 . (3.5) Then w e hav e that the differen tial op erator Λ( X 0 ) tur ns the function g = f u 1 u − 1 to zero: Λ( X 0 ) g ( u 1 , u 0 , u − 1 ) = 0 (3.6) and there is no an y op erator o f low er order whic h ann ulates g . F urther it will b e con v enien t to use the following notation for the commutator of t w o op erators: ad X Y = [ X , Y ]. In terms of this new nota tion, mem b ers of the sequence (3.3) are written a s: W 0 , ad X 0 W 0 , ad 2 X 0 W 0 , . . . , ad k +1 X 0 W 0 . Inte gr a b ility c onditions for two-dim ensional lattic es 7 F orm ula (3.4) takes the form:  D x , ad k X 0 W 0  = − X k 0 ( f u 1 u − 1 ) X 0 . (3.7) Lemma 5. Assume that the cha r acteristic p olynomial Λ( λ ) admits two diffe r ent r o ots λ = α and λ = β . Then either a) α = − β or b) α = − 2 β . Pro of. Let us construct p o lynomials Λ α ( λ ) and Λ β ( λ ) b y the fo llowing rule: Λ α ( λ ) = Λ( λ ) λ − α , Λ β ( λ ) = Λ( λ ) λ − β . Then the op erators P α = Λ α ( ad X 0 W 0 ) , P β = Λ β ( ad X 0 W 0 ) satisfy the relatio ns [ D x , P α ] = A ( u 1 , u − 1 ) e αu 0 X 0 , [ D x , P β ] = B ( u 1 , u − 1 ) e β u 0 X 0 , (3.8) where functions A and B don’t v anish iden tically . These formu las are easily pro v ed, let us b egin t he first one. The op eration of comm utation with D x b y virtue of (3.7) satisfies the formu la: [ D x , Λ α ( ad X 0 W 0 )] = − Λ α ( X 0 ) g X 0 . (3.9) No w w e ha v e to sp ecify the factor g 0 := Λ α ( X 0 ) g , that is a solution of the equation ( X 0 − α ) g 0 = 0. Indeed, since Λ( X 0 ) g = ( X 0 − α )Λ α ( X 0 ) g then w e get the former equation which implies g 0 = A ( u 1 , u − 1 ) e αu 0 , where A ( u 1 , u − 1 ) is a nonzero quasi- p olynomial on u 1 , u − 1 . The second formula of (3 .8) is pro ve d in a similar w a y . W e define a sequence of m ultiple comm utators in suc h a wa y R 1 = [ P α , P β ] , R 2 = [ P α , R 1 ] , ..., R k +1 = [ P α , R k ] , ... Let us ev aluate the comm utator [ D x , R 1 ]. Due to the Jacobi identit y w e hav e [ D x , R 1 ] = [ D x , [ P α , P β ]] = [ P α , [ D x , P β ]] − [ P β , [ D x , P α ]] = =  P α , B ( u 1 , u − 1 ) e β u 0 X 0  − [ P β , A ( u 1 , u − 1 ) e αu 0 X 0 ] . (3.10) By construction the v ector fields P α , P β do not con tain differen tiation with resp ect to the v ariables u 1 , u 0 , u − 1 , therefore (3.1 0 ) implies [ D x , R 1 ] = − B ( u 1 , u − 1 ) e β u 0 [ X 0 , P α ] + A ( u 1 , u − 1 ) e αu 0 [ X 0 , P β ] . (3.11) It remains to ev aluate the comm utators [ X 0 , P α ] and [ X 0 , P β ]. T o this aim find their comm utators with D x : [ D x , [ X 0 , P α ]] = αAe αu 0 P α , [ D x , [ X 0 , P β ]] = β B e β u 0 P β . No w due to (3.8) w e get [ D x , [ X 0 , P α ] − αP α ] = 0 , [ D x , [ X 0 , P β ] − β P β ] = 0 . The la st tw o equations imply in virtue of Lemma 1 the desired relations [ X 0 , P α ] = αP α , [ X 0 , P β ] = β P β . Th us (3.11) give s rise to [ D x , R 1 ] = − αB e β u 0 P α + β Ae αu 0 P β . Inte gr a b ility c onditions for two-dim ensional lattic es 8 By the same wa y w e find [ X 0 , R 1 ] = ( α + β ) R 1 and then deduce the relation [ D x , R 2 ] = ( α + 2 β ) Ae αu 0 R 1 . It can b e prov ed b y induction, that [ D x , R m ] = y m R m − 1 , [ X 0 , R m − 1 ] = z m − 1 R m − 1 , m ≥ 2 , where y k and z k are solutions to the discrete equations y m +1 = y m + Ae αu 0 z m , z m = z m − 1 + α, m ≥ 2 (3.12) with the following initial data z 1 = α + β , y 2 = ( α + 2 β ) Ae αu 0 . (3.13) The problem (3.1 2), (3.13) is solv ed explicitly: z n = α n + β , y n = Ae αu 0 ( n + 1)  n 2 α + β  . Since the c haracteristic algebra is finite dimensional there exists a natural N suc h that R N +1 is linearly expressed through the previous mem b ers of the sequence: R N +1 = λ N R N + λ N − 1 R N − 1 + ...λ 1 R 1 + λ α P α + λ β P β , (3.14) where the op erators R N , R N − 1 , ...R 1 , P α , P β are supp osed to b e linearly independen t. Applying the op erator ad D x to b oth sides of equation (3.14), we find y N +1 R N = D x ( λ N ) R N + λ N y N R N − 1 + ... Collecting co efficie nts b efore R N , w e find the equation D x ( λ N ) = y N +1 . W e concen trate on this equation by represen ting it in an explicit f o rm X j ∂ λ N ∂ u j u j,x + X j ∂ λ N ∂ u j,x u j,xx + ... = y N +1 , where y N +1 = Ae αu 0 ( N + 2 )  N +1 2 α + β  . Since the r.h.s. do es not con tain the v ariables u j,x , u j,xx ,... w e g et immediately that D x ( λ N ) = 0. Hence, this equation is satisfied only when λ N = const and y N +1 = 0, or when N +1 2 α + β = 0. W e can rep eat all the reasoning b y replacing α ↔ β . Then we arrive at a similar relation with some nat ura l K : K +1 2 α + β = 0. In other w ords the following system of equations ( N + 1) α + 2 β = 0 , ( K + 1) β + 2 α = 0 should hav e solution in natural N , K . Solving the system w e get: ( N + 1)( K + 1) = 4 when α β 6 = 0. Note that if αβ = 0 then b ot h o f the r o ots v a nish. How ev er t ha t con tradicts the requiremen t α 6 = β . Th us we hav e either N = K = 1 or K = 0, N = 3. In the first case α = − β , in the second case β = − 2 α . This completes the pro of of Lemma 5. These t w o exceptional cases are studied in the follo wing t heorem. Theorem 2. If the p olynomial (3.5) has two differ ent r o ots α and β = − 2 α (or α and β = − α ) then the char acteristic Lie-Rinehart algebr a c orr es p onding to the r e d uc e d system (1.5) is o f infi nite di m ension. In other words Theorem 2 claimes that in these tw o cases the reduced system is not in tegrable in the sense of D arb oux. Theorem 2 is prov ed in App endix. Inte gr a b ility c onditions for two-dim ensional lattic es 9 3.1. Investigation of multiple r o ots No w let’s study the problem of the m ultiplicit y of the ro o ts o f the p olynomial Λ( λ ) defined b y (3 .5 ). Lemma 6. Polynomial Λ ( λ ) do es not ha v e an y multiple non-zer o r o ot. Pro of. Su pp o se that λ = α is a m ultiple non-zero root of the p olynomial Λ( λ ). Define new p olynomials Λ 1 ( λ ) = 1 λ − α Λ( λ ) and Λ 2 ( λ ) = 1 ( λ − α ) 2 Λ( λ ). Then w e consider the equation Λ 1 ( X ) g ( u 1 , u 0 , u − 1 ) = y . Eviden tly ( X 0 − α ) y = 0, therefore by solving this equation one can find y = A ( u 1 , u − 1 ) e αu 0 . Similarly w e put Λ 2 ( X ) g ( u 1 , u 0 , u − 1 ) = z and then find that ( X 0 − α ) z = Λ 1 ( X ) g = y , whic h implies that z = e αu 0 ( A ( u 1 , u − 1 ) u 0 + B ( u 1 , u − 1 )), where B = B ( u 1 , u − 1 ) is a function. Due to the fo rm ula (3.9) w e can obtain that the o p erators P = Λ 1 ( ad X 0 W 0 ), T = Λ 2 ( ad X 0 W 0 ) satisfy the fo llowing comm utativit y conditions [ D x , P ] = A ( u 1 , u − 1 ) e αu 0 X 0 , [ D x , T ] = e αu 0 ( A ( u 1 , u − 1 ) u 0 + B ( u 1 , u − 1 )) X 0 . Let us construct a sequence of the op erators due to the formulas: K 1 = [ P , T ] , K 2 = [ P , K 1 ] , ..., K m +1 = [ P , K m ] , ... It is easily chec k ed that [ X 0 , P ] = αP , [ X 0 , T ] = αT + P . Indeed, let us c hec k the first of these f orm ulas. Eviden tly we ha v e [ D x , [ X 0 , P ]] = [ X 0 , Ae αu 0 X 0 ] = αAe αu 0 X 0 . Therefore, [ D x , [ X 0 , P ] − αP ] = 0 . By virtue of Lemma 1 one o bta ins the f o rm ula desired. In a similar wa y w e prov e that [ D x , K 1 ] = e αu ( αAu + αB − A ) P − e αu 0 αAT , [ D x , K 2 ] = 3 α Ae αu 0 K 1 , [ X 0 , K 1 ] = 2 αK 1 , [ X 0 , K 2 ] = 3 αK 2 . It can b e prov ed b y induction that f o r an y m ≥ 2 [ D x , K m ] = α 2 ( m + 1) mAe αu 0 K m − 1 . Since the Lie-R inehart algebra generated by P , T is supp osed to b e of a finite dimension then there is an in teger M suc h that K M +1 = a M K M + a M − 1 K M − 1 + ... + a 1 K 1 + b 1 P + b 2 T , (3.15) where a j , b j are some functions dep ending on the dynamical v ariables u j , u j x , u j xx , ... and the op erat o rs K M , K M − 1 , ...K 1 , P , T are linearly indep enden t. By applying the op erator ad D x to equation (3 .15) one gets for M > 0 1 2 α ( M + 2)( M + 1) Ae αu 0 K M = D x ( a M ) K M + ..., where the tail contains the summands with K M − 1 , K M − 2 , ... . Th us the last equation implies D x ( a M ( ¯ u, ¯ u x , ¯ u xx , ... )) = α 2 ( M + 2)( M + 1) A ( u 1 , u − 1 ) e αu 0 . (3.16) Inte gr a b ility c onditions for two-dim ensional lattic es 10 Equation (3.16) yields D x ( a M ) = 0 and α = 0 . The latter contradicts the assumption α 6 = 0. The case M = 0 i.e. K 1 = b 1 P + b 2 T sh ould b e inv estigated separately . Here application of the o p erator ad D x yields e αu 0 ( αAu + αB − A ) P − αe αu 0 AT = = D x ( b 1 ) P + D x ( b 2 ) T + b 1 Ae αu 0 X 0 + b 2 e αu 0 ( Au 0 + B ) X 0 . Comparison o f the co efficien ts b efore t he op erators X 0 , P , T shows that this equation is con tradictory . This completes the pro of of Lemma 6. Lemma 7. At the p oint λ = 0 p olynomial Λ( λ ) defi n e d i n (3 . 5) might have only a simple r o o t. Pro of. Assume that α = 0 is a ro ot the c haracteristic p olynomial Λ( λ ) of the m ultiplicit y k . Then then due to Theorem 2 w e hav e Λ( λ ) = λ k . Let us construct op erators: P 1 = Λ 1 ( ad X 0 W 0 ) = ad k − 1 X 0 W 0 , P 2 = Λ 2 ( ad X 0 W 0 ) = ad k − 2 X 0 W 0 . It can b e prov ed that they satisfy relations: [ D x , P 1 ] = − A ( u 1 , u − 1 ) X 0 , [ D x , P 2 ] = − ( A ( u 1 , u − 1 ) u 0 + B ( u 1 , u − 1 )) X 0 . Let us prov e that the L ie- Rinehart algebra g enerated by P 1 and P 2 , where the co efficien t A ( u 1 , u − 1 ) do esn’t v a nish iden tically , is of infinite dimension. Define a sequence of the multiple commutators in suc h a w ay P 1 , P 2 , P 3 = [ P 2 , P 1 ] , ..., P m = [ P 2 , P m − 1 ] , ... It can b e prov ed b y induction on m that [ D x , P m ] = A ( u 1 , u − 1 ) P m − 1 . If the Lie- Rinehart algebra is of a finite dimension then there exists a natural M suc h that P M +1 = µ M P M + µ M − 1 P M − 1 + ... + µ 1 P 1 . By applying the op erator D x to b oth sides of the last equalit y , w e obta in: A ( u 1 , u − 1 ) P M = D x ( µ M ) P M + µ M A ( u 1 , u − 1 ) P M + ... + + D x ( µ 1 ) P 1 + µ 1 A ( u 1 , u − 1 ) P 1 . By comparing the co efficien ts b efore P M w e get: A ( u 1 , u − 1 ) = D x ( µ M ) . (3.17) Since µ M = µ M ( ¯ u, ¯ u x , ¯ u xx , ... ) dep ends on a set of the dynamical v ariable while A dep ends o nly on u 1 and u − 1 equalit y (3.17) fails to b e true unless A ( u 1 , u − 1 ) = 0 that con tradicts to our assumption. Therefore t he L ie- Rinehart a lg ebra generated b y the op erators P 1 and P 2 is of infinite dimension. Lemma 7 is prov ed. Inte gr a b ility c onditions for two-dim ensional lattic es 11 Th us, summarizing the statemen ts of Lemmas 5–7, w e conclude t ha t the p olynomial Λ( λ ) defined b y ( 3 .5) migh t hav e only one ro ot and this ro ot is simple. Therefore, equation (3.6) has the form ( X 0 − α ) f u 1 u − 1 = 0, where α is a constant. Th us, w e ha v e that the function f u 1 u − 1 has the following fo rm: f u 1 u − 1 = Q ( u 1 , u − 1 ) e αu 0 , where Q ( u 1 u − 1 ) is a function b eing a quasi-p olynomial with resp ect to an y of it s argumen ts u 1 , u − 1 . No w, let us rep eat the reasoning a b o v e by c hanging the op erator X 0 b y X 1 . F or this purp ose, w e construct a sequence as follows H 0 = W 0 , H 1 = [ X 1 , H 0 ] , H 2 = [ X 1 , H 1 ] , . . . , H k +1 = [ X 1 , H k ] , . . . Elemen ts o f t he sequence satisfy the relations: [ D x , H k ] = − X k 1 ( f u 1 u − 1 ) X 0 = − e αu 0 X k 1 ( Q u 1 u − 1 ) X 0 . Since the c haracteristic algebra is finite-dimensional there exists a natura l K suc h that H K +1 is linearly expressed b y the previous mem b ers: H K +1 + λ K H K + · · · + λ 1 H 1 + λ 0 H 0 = 0 , where the op erators H 0 , H 1 , . . . , H k are linearly indep enden t. W e apply the op era t o r ad D x to the obtained equation and g et relations D x ( λ j ) = 0 for j = 0 , 1 , ...K and also a relation Ω( X 1 ) Q u 1 u − 1 = 0, where Ω( λ ) = λ K +1 + λ K λ K + · · · + λ 0 (3.18) is a quasi-p olynomial with constan t co efficien ts. No w w e inv es tigat e the c haracteristic p o lynomial (3.18) b y using the reasonings w e a pplied ab o v e to t he characteris tic p o lynomial (3 .5). As a result w e prov e that Q u 1 u − 1 = Φ( u − 1 ) e β u 1 and f u 1 u − 1 = Φ( u − 1 ) e αu 0 + β u 1 . Here Φ( u − 1 ) is a quasi-p olynomial. Finally , w e rep eat this reasoning, replacing X 1 b y X − 1 and pro v e t he fo llo wing statemen t: Theorem 3. If l a ttic e (1.1) i s inte gr able in the sen s e of Defi nition 1 then the function f ( u n +1 , u n , u n − 1 ) satisfies the fol lowin g e quation: f u n +1 ,u n − 1 = C e αu n + β u n +1 + γ u n − 1 , (3.19) wher e C , α , β , γ ar e c onstant. 4. The necessary in tegrability c onditions Theorem 4. If α = 0 , C 6 = 0 in (3.19) then β = 0 and γ = 0 . If α 6 = 0 , C 6 = 0 then β = − α 2 m , γ = − α 2 k , wher e m , k a r e nonne gative inte gers. Pro of. Supp ose that C 6 = 0 then a minimal order op erator (3.6) whic h ann ulates the function g = f u 1 u − 1 has the fo r m Λ( X 0 ) = X 0 − α . Let us construct the op erator P 0 = Λ( ad X 0 W 0 ) /C ( λ − α ) and the op erat o r P 1 = D n P 0 D − 1 n , where D n stands for the Inte gr a b ility c onditions for two-dim ensional lattic es 12 shift op erator acting as D n y ( n ) = y ( n + 1). Th us we hav e the o p erators P 0 , P 1 ∈ L of the form P j = X k a j k (1) ∂ ∂ u k x + a j k (2) ∂ ∂ u k xx + a j k (3) ∂ ∂ u k xxx + ... Due to (3.19) and Lemma 4 these op erator s satisfy the following comm utativit y relations [ D x , P 0 ] = e ω X 0 , [ D x , P 1 ] = e ω 1 X 1 , (4.1) where ω = αu 0 + β u 1 + γ u − 1 , ω 1 = αu 1 + β u 2 + γ u 0 . The first relation in (4 .1) is easily pro v ed by the same w ay as form ula (3 .8). The second relation follows from the first one b y applying the conjugation tra nsfor ma t io n X → D n X D − 1 n . One can easily v erify that [ X 0 , P 0 ] = αP 0 , [ X 0 , P 1 ] = γ P 1 , [ X 1 , P 0 ] = β P 0 , [ X 1 , P 1 ] = αP 1 . Let us construct a sequence of the multiple commutators as follows K 1 = [ P 0 , P 1 ] , K 2 = [ P 0 , K 1 ] , ..., K m = [ P 0 , K m − 1 ] , ... It is c hec k ed by direct calculation that [ D x , K 1 ] = − β e ω 1 P 0 + γ e ω P 1 , [ X 0 , K 1 ] = ( α + γ ) K 1 , [ D x , K 2 ] = ( α + 2 γ ) e ω K 1 , [ X 0 , K 2 ] = (2 α + γ ) K 2 . By induction we can prov e that for n ≥ 2 [ D x , K n ] = e ω π n K n − 1 , [ X 0 , K n ] = y n K n (4.2) with π n = α 2 n 2 + ( γ − α 2 ) n , y n = αn + γ . Since the characteristic algebra is of a finite dimension then there exists a natural N suc h that K N +1 = λ N K N + ... + λ 1 K 1 + µ 0 P 0 + µ 1 P 1 , (4.3) where the op erato rs P 0 , P 1 , K 1 ,..., K N are linearly indep enden t. By applying to (4.3) the op erator ad D x one find due to (4.2) that e ω π N +1 K N = D x ( λ N ) K N + λ N e ω π N K N − 1 + ... (4.4) W e compare the co efficien ts b efore K N in (4.4) a nd w e get λ N = 0, π N +1 = 0. Or the same α 2 N + γ = 0 . (4.5) F rom this form ula it follows that if α = 0 then γ = 0. Similarly one can prov e tha t if α = 0 then β = 0. The case N = 0 is neve r realized. Indeed supp osing K 1 = µ 0 P 0 + µ 1 P 1 one o btains a con tradictory equation − β e ω 1 P 0 + γ e ω P 1 = D x ( µ 0 ) P 0 + D x ( µ 1 ) P 1 + µ 0 e ω X 0 + µ 1 e ω 1 X 1 unless β = 0 , γ = 0. Inte gr a b ility c onditions for two-dim ensional lattic es 13 Th us it follow s from (4.5) that γ = − α 2 N . F orm ula β = − α 2 m is prov ed in a similar w a y . Moreov er, w e see that if γ = 0 t hen β = 0. And similarly w e obtain that if β = 0 then γ = 0. In other w ords, if β γ 6 = 0 then f u 1 u − 1 has the following fo rm f u 1 u − 1 = C e αu 0 − αm 2 u 1 − αk 2 u − 1 , where C 6 = 0, α 6 = 0. If β = 0 or γ = 0 then b oth β = γ = 0 and f u 1 u − 1 = C e αu 0 , where α 6 = 0, C 6 = 0. Theorem 4 is pro v ed. The main result is giv en in Theorem 5. L attic e (1.1) which is inte gr ab l e in the sense of D efinition 1, c an b e r e duc e d by suitable r es c alings to on e of the fol lowing fo rm s: u n,xy = e αu n − α 2 mu n +1 − α 2 k u n − 1 + a ( u n +1 , u n ) + b ( u n , u n − 1 ) , u n,xy = e αu n u n +1 u n − 1 + a ( u n +1 , u n ) + b ( u n , u n − 1 ) , u n,xy = u n +1 u n − 1 + a ( u n +1 , u n ) + b ( u n , u n − 1 ) , u n,xy = a ( u n +1 , u n ) + b ( u n , u n − 1 ); her e α 6 = 0 and m , k ar e p ositive inte gers. Theorem 5 straightforw ardly follows from Theorems 3, 4. 5. App endix Here we giv e a complete pro of of Theorem 2. W e consider the cases β = − 2 α and β = − α separately . Our pro of uses the sche me applied earlier in [2 3], [24]. 5.1. The c ase β = − 2 α In this subsection w e will prov e tha t if p o lynomial Λ( λ ) has t w o differen t nonzero ro ots α and β = − 2 α then the Lie-Rinehart algebra L generated b y the op erators X 0 and W 0 is of an infinite dimension. First w e intro duce t wo p o lynomials according to the rule Λ α ( λ ) = Λ( λ ) λ − α , Λ β ( λ ) = Λ( λ ) λ + 2 α . Then w e construct t w o op erators P α , P β ∈ L : P α = Λ α ( ad X 0 W 0 ) , P β = Λ β ( ad X 0 W 0 ) and concen trate on the Lie-Rinehart a lg ebra L 1 ⊂ L b eing a subalgebra of L generated b y the op erators P α , P β . By construction these op erators satisfy the follow ing comm utativit y relations [ D x , P α ] = A ( u 1 , u − 1 ) e αu 0 X 0 , [ D x , P β ] = B ( u 1 , u − 1 ) e − 2 αu 0 X 0 , where A ( u 1 , u − 1 ), B ( u 1 , u − 1 ) ar e quasi-p olynomials in u 1 , u − 1 . W e assume tha t ( u − 1 , u 1 ) ∈ D , where D is a domain in C 2 , where b oth A , B do not v anish. Inte gr a b ility c onditions for two-dim ensional lattic es 14 Let us consider the op erators: Y 1 = P α + P β , Y 2 = ∂ ∂ u 0 . (5.1) Assume that L i stands for the linear space spanned by all p ossible commutators of the op erators Y 1 and Y 2 of the length less or equal to i − 1, where i = 2 , 3 , . . . . W e emphasize that the linear com bination in this space is tak en with co efficien ts b eing the functions depending on a finite n um b er of the v ariables ¯ u , ¯ u x , ¯ u xx , . . . . Th us L 2 = { Y 1 , Y 2 } is the linear span of Y 1 and Y 2 , dim L 2 = 2 . Similarly L 3 is the linear en v elop o f the vec tor Y 1 , Y 2 and Y 3 = [ Y 1 , Y 2 ], i.e. L 3 = { Y 1 , Y 2 , Y 3 } . The refore, L 4 = { Y 1 , Y 2 , Y 3 , [ Y 1 , Y 3 ] , [ Y 2 , Y 3 ] } etc. Let us denote δ ( i ) = dim L i − dim L i − 1 . W e also will use the following notations for m ultiple comm utators: Y i 1 ,...,i n = ad Y i 1 . . . ad Y i n − 1 Y i n , where ad Y W = [ Y , W ] . Lemma 8. Assume that p olynomial Λ( λ ) has two differ ent non zer o r o ots α and − 2 α . Then the f o l lowing formulas ar e true: δ ( i ) = 2 , i = 6 n + 2 , i = 6 n + 4 , n = 1 , 2 , . . . , δ ( i ) = 1 , i = 6 n − 1 , i = 6 n, i = 6 n + 1 , i = 6 n + 3 , n = 1 , 2 , . . . , L 6 n +2 = L 6 n +1 ⊕ { Y 1 ... 121 , Y 21 ... 121 } , L 6 n +4 = L 6 n +3 ⊕ { Y 1 ... 121 , Y 21 ... 121 } , L 6 n − 1 = L 6 n − 2 ⊕ { Y 1 ... 121 } , L 6 n = L 6 n − 1 ⊕ { Y 1 ... 121 } , L 6 n +1 = L 6 n ⊕ { Y 1 ... 121 } , L 6 n +3 = L 6 n +2 ⊕ { Y 1 ... 121 } . Pro of. W e intro duced the op erators Y 1 , Y 2 b y f orm ulas (5.1). The followin g comm utation relatio ns are true fo r t hese op erators: [ D x , Y 1 ] =  Ae αu 0 + B e − 2 αu 0  Y 2 , [ D x , Y 2 ] = 0 . (5.2) W e in tro duce the op erat o r of length 2: Y 3 = [ Y 2 , Y 1 ] = Y 21 . Then using the Jacobi iden tit y and formulas (5.2 ) , we get [ D x , Y 3 ] = α  Ae αu 0 − 2 B e − 2 αu 0  Y 2 . (5.3) If w e assume t hat Y 3 is linearly expressed b y Y 1 and Y 2 , i.e. Y 3 = λ 1 Y 1 + λ 2 Y 2 , (5.4) then w e get a con tradiction. Indeed by commu ting b oth sides of (5.4) with D x and then simplifying due to (5.2), (5.3) we obtain α  Ae αu 0 − 2 B e − 2 αu 0  Y 2 = D x ( λ 1 ) Y 1 + λ 1  Ae αu 0 + B e − 2 αu 0  Y 2 + D x ( λ 2 ) Y 2 . Inte gr a b ility c onditions for two-dim ensional lattic es 15 Comparing the co efficien ts b efore indep enden t o p erators Y 1 , Y 2 , w e get: D x ( λ 1 ) = 0 a nd α  Ae αu 0 − 2 B e − 2 αu 0  = λ 1  Ae αu 0 + B e − 2 αu 0  + D x ( λ 2 ) . The last equality implies that D x ( λ 2 ) = 0 and λ 1 − α = 0, λ 1 + 2 α = 0. Ob viously a pair of these equations is inconsisten t b ecause α 6 = 0. W e in tro duce the comm utators of length 3: Y 4 = [ Y 1 , Y 3 ] and ¯ Y 4 = [ Y 2 , Y 3 ]. Then  D x , ¯ Y 4  = α 2  Ae αu 0 + 4 B e − 2 αu 0  Y 2 , (5.5) [ D x , Y 4 ] = − α  2 Ae αu 0 − B e − 2 αu 0  Y 3 + 2 α 2  Ae αu 0 + B e − 2 αu 0  Y 1 . One can see that  D x , ¯ Y 4  = 2 α 2 [ D x , Y 1 ] − α [ D x , Y 3 ] = [ D x , 2 α 2 Y 1 − αY 3 ]. D ue to Lemma 1 this equalit y implies that ¯ Y 4 = 2 α 2 Y 1 − αY 3 . The op erator Y 4 = Y 121 is not linearly expressed t hro ugh t he op erators of low er order. Thus , we ha v e L 4 = { Y 1 , Y 2 , Y 3 , Y 4 } . W e in tro duce the comm utator s o f lengh t 4: Y 5 = [ Y 1 , Y 4 ] and ¯ Y 5 = [ Y 2 , Y 4 ]. Using the Jacobi iden tity and formulas (5.2) , ( 5 .5), w e find  D x , ¯ Y 5  = α 2  2 Ae αu 0 − B e − 2 αu 0  Y 3 − 2 α 3  Ae αu 0 + B e − 2 αu 0  Y 1 = [ D x , − αY 4 ] , [ D x , Y 5 ] = − α  2 Ae αu 0 − B e − 2 αu 0  Y 4 −  Ae αu 0 + B e − 2 αu 0  [ Y 4 , Y 2 ] = − 3 αAe αu 0 Y 4 . Due t o Lemma 1, we conclude that ¯ Y 5 = − α Y 4 . Th e o p erator Y 5 = Y 1121 is not linearly expressed through the comm utators of low er order. Th us, w e hav e L 5 = { Y 1 , Y 2 , Y 3 , Y 4 , Y 5 } . Let us consider the comm utators of the length 5 : Y 6 = [ Y 1 , Y 5 ] , ¯ Y 6 = [ Y 2 , Y 5 ] , [ Y 3 , Y 4 ] The following formulas a r e true:  D x , ¯ Y 6  = 0 , [ D x , [ Y 3 , Y 4 ]] = − 3 α 2 Ae αu 0 Y 4 = [ D x , αY 5 ] , [ D x , Y 6 ] = − 3 αAe αu 0 Y 5 . Using Lemma 1, w e conclude that [ Y 3 , Y 4 ] = αY 5 . The op erat or Y 6 = Y 11121 is not linearly expresse d by the op erator s of low er order. So, w e hav e L 6 = { Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , Y 6 } . No w w e in tro duce the op erators of the length 6: Y 7 = [ Y 1 , Y 6 ] = Y 111121 , ¯ Y 7 = [ Y 2 , Y 6 ] = Y 211121 , [ Y 3 , Y 5 ] . One can prov e that the following fo rm ulas are tr ue: ¯ Y 7 = αY 6 , [ Y 3 , Y 5 ] = αY 6 , [ D x , Y 7 ] = α  − 2 Ae αu 0 + B e − 2 αu 0  Y 6 . The op erato r Y 7 = Y 111121 is not linearly expressed b y the op erators of lo w er order and L 7 = { Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , Y 6 , Y 7 } . Then w e consider the op erato r s of the length 7: Y 8 = [ Y 1 , Y 7 ] , ¯ Y 8 = [ Y 2 , Y 7 ] , [ Y 3 , Y 6 ] , [ Y 4 , Y 5 ] . One can prov e that  D x , ¯ Y 8  = α 2  − 4 Ae αu 0 − B e − 2 αu 0  Y 6 , [ D x , Y 8 ] = α  − 2 Ae αu 0 + B e − 2 αu 0  Y 7 +  Ae αu 0 + B e − 2 αu 0  ¯ Y 8 . Inte gr a b ility c onditions for two-dim ensional lattic es 16 The op erators Y 8 , ¯ Y 8 is not expressed through the op erato rs of lo w er order. It is not difficult to sho w that [ Y 3 , Y 6 ] = ¯ Y 8 − α Y 7 , [ Y 4 , Y 5 ] = 2 αY 7 − ¯ Y 8 . Thus we hav e that the space L 8 is o bt a ined fr o m L 7 b y adding t wo linearly indep enden t elemen ts Y 8 = Y 1111121 and ¯ Y 8 = Y 2111121 , i.e. L 8 =  Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , Y 6 , Y 7 , Y 8 , ¯ Y 8  . No w let us intro duce the op erators of the length 8: Y 9 = [ Y 1 , Y 8 ] , ¯ Y 9 = [ Y 2 , Y 8 ] ,  Y 1 , ¯ Y 8  ,  Y 2 , ¯ Y 8  , [ Y 3 , Y 7 ] , [ Y 4 , Y 6 ] . One can show that [ Y 4 , Y 6 ] = αY 8 , [ Y 3 , Y 7 ] = − αY 8 ,  Y 2 , ¯ Y 8  = 2 α 2 Y 7 + α ¯ Y 8 ,  Y 1 , ¯ Y 8  = αY 8 , ¯ Y 9 = 0 , [ D x , Y 9 ] = α  − Ae αu 0 + 2 B e − 2 αu 0  Y 8 . So w e see that Y 9 = Y 11111121 is not linearly expressed through the op erators of low er order and L 9 = L 8 ⊕ Y 9 . The op erato rs of length 9 are constructed b y t he follow ing w a y: Y 10 = [ Y 1 , Y 9 ] , ¯ Y 10 = [ Y 2 , Y 9 ] ,  Y 3 , ¯ Y 8  , [ Y 3 , Y 8 ] , [ Y 4 , Y 7 ] , [ Y 5 , Y 6 ] . F or these op erato rs the relations hold:  Y 3 , ¯ Y 8  = − 3 α 2 Y 8 , [ Y 3 , Y 8 ] = ¯ Y 10 , [ Y 4 , Y 7 ] = − αY 9 − ¯ Y 10 , [ Y 5 , Y 6 ] = 2 αY 9 + ¯ Y 10 , [ D x , Y 10 ] = α  − Ae αu 0 + 2 B e − 2 αu 0  Y 9 +  Ae αu 0 + B e − 2 αu 0  ¯ Y 10 ,  D x , ¯ Y 10  = α 2  − Ae αu 0 − 4 B e − 2 αu 0  Y 8 . The op erators Y 10 = Y 11111112 1 and ¯ Y 10 = X 21111112 1 are not linearly expresse d through the op erators of low er order, L 10 = L 9 ⊕  Y 10 , ¯ Y 10  . No w let us intro duce the no t ation: Y n = [ Y 1 , Y n − 1 ], ¯ Y n = [ Y 2 , Y n − 1 ]. W e pro v e Lemma 7 b y induction. Assum e that f o r i = n − 1 the following formulas ar e true:  D x , Y 6( n − 1) − 1  = − α  2 Ae αu 0 − B e − 2 αu 0  Y 6( n − 1) − 2 + (5.6)  Ae αu 0 + B e − 2 αu 0  Y 2 , Y 6( n − 1) − 2  ,  D x , Y 6( n − 1)  = − 3 αAe αu 0 Y 6( n − 1) − 1 +  Ae αu 0 + B e − 2 αu 0   Y 2 , Y 6( n − 1) − 1  , (5.7)  D x , Y 6( n − 1)+1  = − 3 αAe αu 0 Y 6( n − 1) +  Ae αu 0 + B e − 2 αu 0   Y 2 , Y 6( n − 1)  , (5.8)  D x , Y 6( n − 1)+2  = α  − 2 Ae αu 0 + B e − 2 αu 0  Y 6( n − 1)+1 + +  Ae αu 0 + B e − 2 αu 0   Y 2 , Y 6( n − 1)+1  , ( 5 .9)  D x , Y 6( n − 1)+3  = α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6( n − 1)+2 + +  Ae αu 0 + B e − 2 αu 0   Y 2 , Y 6( n − 1)+2  , (5.10)  D x , Y 6( n − 1)+4  = α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6( n − 1)+3 + +  Ae αu 0 + B e − 2 αu 0   Y 2 , Y 6( n − 1)+3  , (5.11) ¯ Y 6( n − 1) =  Y 2 , Y 6( n − 1) − 1  = 0 , (5.12) ¯ Y 6( n − 1) − 1 = − α Y 6( n − 1) − 2 , (5.13) ¯ Y 6( n − 1)+1 = α Y 6( n − 1) , (5.14) Inte gr a b ility c onditions for two-dim ensional lattic es 17 ¯ Y 6( n − 1)+3 = 0 , (5.15)  Y 1 , ¯ Y 6( n − 1)+2  = α Y 6( n − 1)+2 , (5.16)  Y 2 , ¯ Y 6( n − 1)+2  = 2 α 2 Y 6( n − 1)+1 + α ¯ Y 6( n − 1)+2 , (5.17)  Y 1 , ¯ Y 6( n − 1)+4  = − α Y 6( n − 1)+4 , (5.18)  Y 2 , ¯ Y 6( n − 1)+4  = 2 α 2 Y 6( n − 1)+3 − α ¯ Y 6( n − 1)+4 . (5.19) Let us pro ve form ulas (5 .6)–(5.19) for i = n . W e in tro duce the commutators of length 6 n − 2: Y 6 n − 1 = Y 6( n − 1)+5 =  Y 1 , Y 6( n − 1)+4  , ¯ Y 6 n − 1 = ¯ Y 6( n − 1)+5 =  Y 2 , Y 6( n − 1)+4  . Using the Ja cobi iden tity and fo rm ulas (5.6)–(5.19) we find that the following equalities hold:  D x , ¯ Y 6 n − 1  = α 2  Ae αu 0 − 2 B e − 2 αu 0  Y 6( n − 1)+3 − α  Ae αu 0 + B e − 2 αu 0  ¯ Y 6( n − 1)+4 = =  D x , − αY 6( n − 1)+4  , [ D x , Y 6 n − 1 ] =  D x ,  Y 1 , Y 6( n − 1)+4  =  Y 1 ,  D x , Y 6( n − 1)+4  −  Y 6( n − 1)+4 , [ D x , Y 1 ]  = =  Y 1 , α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6( n − 1)+3 +  Ae αu 0 + B e − 2 αu 0   Y 2 , Y 6( n − 1)+3  − −  Y 6( n − 1)+4 ,  Ae αu 0 + B e − 2 αu 0  Y 2  = α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6( n − 1)+4 + +  Ae αu 0 + B e − 2 αu 0   Y 1 ,  Y 2 , Y 6( n − 1)+3  +  Ae αu 0 + B e − 2 αu 0   Y 2 , Y 6( n − 1)+4  = = α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6( n − 1)+4 +  Ae αu 0 + B e − 2 αu 0   Y 2 , ¯ Y 6( n − 1)+4  + +  Ae αu 0 + B e − 2 αu 0   Y 2 , Y 6( n − 1)+4  = − 3 α Ae αu 0 Y 6( n − 1)+4 = − 3 αAe αu 0 Y 6 n − 2 . Using Lemma 1 w e conclude that ¯ Y 6 n − 1 = − α Y 6( n − 1)+4 . One can see that the op erato r Y 6 n − 1 is not linearly expressed thro ug h the op erators of less indices. Thus we obtain L 6 n − 1 = L 6 n − 2 ⊕ { Y 6 n − 1 } , δ (6 n − 1) = 1. No w w e consider the commutators of length 6 n − 1: Y 6 n = [ Y 1 , Y 6 n − 1 ] , ¯ Y 6 n = [ Y 2 , Y 6 n − 1 ] . The formulas of commutation with the op erator D x are:  D x , ¯ Y 6 n  = [ D x , [ Y 2 , Y 6 n − 1 ]] = = [ Y 2 , − 3 αAe αu 0 Y 6 n − 2 ] = − 3 α 2 Ae αu 0 Y 6 n − 2 − 3 αAe αu 0 [ Y 2 , Y 6 n − 2 ] = = − 3 Aα 2 e αu 0 Y 6 n − 2 + 3 α 2 Ae αu 0 Y 6( n − 1)+4 = 0 , [ D x , Y 6 n ] = [ D x , [ Y 1 , Y 6 n − 1 ]] = [ Y 1 , − 3 αAe αu 0 Y 6 n − 2 ] − −  Y 6 n − 1 ,  Ae αu 0 + B e − 2 αu 0  Y 2  = − 3 αAe αu 0 Y 6 n − 1 + +  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n = − 3 α Ae αu 0 Y 6 n − 1 . According to Lemma 1, we conclude that ¯ Y 6 n = 0. The op erator Y 6 n is no t linearly expresse d through the op erators of low er order. The equalities L 6 n = L 6 n − 1 ⊕ { Y 6 n } , δ (6 n ) = 1 a r e true. Let us consider the comm utators of length 6 n : Y 6 n +1 = [ Y 1 , Y 6 n ] , ¯ Y 6 n +1 = [ Y 2 , Y 6 n ] . Inte gr a b ility c onditions for two-dim ensional lattic es 18 The following formulas hold:  D x , ¯ Y 6 n +1  = [ D x , [ Y 2 , Y 6 n ]] = [ Y 2 , − 3 αAe αu 0 Y 6 n − 1 ] = = − 3 α 2 Ae αu 0 Y 6 n − 1 − 3 αAe αu 0 [ Y 2 , Y 6 n − 1 ] = − 3 α 2 Ae αu 0 Y 6 n − 1 = [ D x , αY 6 n ] , [ D x , Y 6 n +1 ] = [ D x , [ Y 1 , Y 6 n ]] = [ Y 1 , − 3 αAe αu 0 Y 6 n − 1 ] −  Y 6 n ,  Ae αu 0 + B e − 2 αu 0  Y 2  = = − 3 α Ae αu 0 Y 6 n +  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +1 = α  − 2 Ae αu 0 + B e − 2 αu 0  Y 6 n . So we ha ve  D x , ¯ Y 6 n +1  = [ D x , αY 6 n ]. Due to Lemma 1 the last equalit y implies that ¯ Y 6 n +1 = α Y 6 n . The op erator Y 6 n +1 is not linearly expressed through the op erator s o f lo w er order, L 6 n +1 = L 6 n ⊕ { Y 6 n +1 } , δ (6 n + 1) = 1. Let us consider the comm utators of length 6 n + 1: Y 6 n +2 = [ Y 1 , Y 6 n +1 ] , ¯ Y 6 n +2 = [ Y 2 , Y 6 n +1 ] . F or these op erato rs the follo wing form ulas are satisfied:  D x , ¯ Y 6 n +2  = [ D x , [ Y 1 , Y 6 n +1 ]] =  Y 1 , α  − 2 Ae αu 0 + B e − 2 αu 0  Y 6 n  − −  Y 6 n +1 ,  Ae αu 0 + B e − 2 αu 0  Y 2  = α  − 2 Ae αu 0 + B e − 2 αu 0  Y 6 n +1 + +  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +2 . The op erators Y 6 n +2 = [ Y 1 , Y 6 n +1 ] and ¯ Y 6 n +2 = [ Y 2 , Y 6 n +1 ] are not linearly expressed b y op erators of lo w er order. The equalities are true: L 6 n +2 = L 6 n +1 ⊕  Y 6 n +2 , ¯ Y 6 n +2  , δ (6 n + 2) = 2. W e in tro duce the comm utators of length 6 n + 2: Y 6 n +3 = [ Y 1 , Y 6 n +2 ] , ¯ Y 6 n +3 = [ Y 2 , Y 6 n +2 ] ,  Y 1 , ¯ Y 6 n +2  ,  Y 2 , ¯ Y 6 n +2  . The following formulas a r e satisfied:  D x ,  Y 2 , ¯ Y 6 n +2  =  Y 2 , α 2  − 4 Ae αu 0 − B e − 2 αu 0  Y 6 n  = = α 3  − 4 Ae αu 0 + 2 B e − 2 αu 0  Y 6 n + α 2  − 4 Ae αu 0 − B e − 2 αu 0  [ Y 2 , Y 6 n ] = = α 3  − 4 Ae αu 0 + 2 B e − 2 αu 0  Y 6 n + α 3  − 4 Ae αu 0 − B e − 2 αu 0  Y 6 n = = α 3  − 8 Ae αu 0 + B e − 2 αu 0  Y 6 n =  D x , 2 α 2 Y 6 n +1 + α ¯ Y 6 n +2  . So w e ha v e that  D x ,  Y 2 , ¯ Y 6 n +2  =  D x , 2 α 2 Y 6 n +1 + α ¯ Y 6 n +2  . Apply Lemma 1 to this equalit y we conclude that  Y 2 , ¯ Y 6 n +2  = 2 α 2 Y 6 n +1 + α ¯ Y 6 n +2 . Then, w e find  D x ,  Y 1 , ¯ Y 6 n +2  =  Y 1 , α 2  − 4 Ae αu 0 − B e − 2 αu 0  Y 6 n  −  ¯ Y 6 n +2 ,  Ae αu 0 + B e − 2 αu 0  Y 2  = = α 2  − 4 Ae αu 0 − B e − 2 αu 0  Y 6 n +1 +  Ae αu 0 + B e − 2 αu 0   Y 2 , ¯ Y 6 n +2  = = α 2  − 4 Ae αu 0 − B e − 2 αu 0  Y 6 n +1 +  Ae αu 0 + B e − 2 αu 0   2 α 2 Y 6 n +1 + α ¯ Y 6 n +2  = = α 2  − 2 Ae αu 0 + B e − 2 αu 0  Y 6 n +1 + α  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +2 = = [ D x , αY 6 n +2 ] . Th us we ha ve  D x ,  Y 1 , ¯ Y 6 n +2  − α Y 6 n +2  . Due to Lemma 1 this form ula g iv es  Y 1 , ¯ Y 6 n +2  = αY 6 n +2 . Then we find  D x , ¯ Y 6 n +3  = [ D x , [ Y 2 , Y 6 n +2 ]] = Inte gr a b ility c onditions for two-dim ensional lattic es 19 =  Y 2 , α  − 2 Ae αu 0 + B e − 2 αu 0  Y 6 n +1 +  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +2  = = α 2  − 2 Ae αu 0 − 2 B e − 2 αu 0  Y 6 n +1 + α  − 2 Ae αu 0 + B e − 2 αu 0  [ Y 2 , Y 6 n +1 ] + + α  Ae αu 0 − 2 B e − 2 αu 0  ¯ Y 6 n +2 +  Ae αu 0 + B e − 2 αu 0   Y 2 , ¯ Y 6 n +2  = α 2  − 2 Ae αu 0 − 2 B e − 2 αu 0  Y 6 n +1 + α  − 2 Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +2 + + α  Ae αu 0 − 2 B e − 2 αu 0  ¯ Y 6 n +2 +  Ae αu 0 + B e − 2 αu 0   2 α 2 Y 6 n +1 + α ¯ Y 6 n +2  = 0 . It is clear by Lemma 1 that ¯ Y 6 n +3 = 0 . Now we calculate [ D x , Y 6 n +3 ] = [ D x , [ Y 1 , Y 6 n +2 ]] =  Y 1 , α  − 2 Ae αu 0 + B e − 2 αu 0  Y 6 n +1 +  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +2  − −  Y 6 n +2 ,  Ae αu 0 + B e − 2 αu 0  Y 2  = α  − 2 Ae αu 0 + B e − 2 αu 0  Y 6 n +2 + +  Ae αu 0 + B e − 2 αu 0   Y 1 , ¯ Y 6 n +2  +  Ae αu 0 + B e − 2 αu 0  [ Y 2 , Y 6 n +2 ] = = α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6 n +2 . A t this step w e obtain L 6 n +3 = L 6 n +2 ⊕ { Y 6 n +3 } , δ (6 n + 1) = 1. Let us consider the comm utators of length 6 n + 3: Y 6 n +4 = [ Y 1 , Y 6 n +3 ] , ¯ Y 6 n +4 = [ Y 2 , Y 6 n +3 ] . The following formulas a r e satisfied:  D x , ¯ Y 6 n +4  = [ D x , [ Y 2 , Y 6 n +3 ]] =  Y 2 , α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6 n +2  = = α 2  − Ae αu 0 − 4 B e − 2 αu 0  Y 6 n +2 + α  − Ae αu 0 + 2 B e − 2 αu 0  [ Y 2 , Y 6 n +2 ] = = α 2  − Ae αu 0 − 4 B e − 2 αu 0  Y 6 n +2 . [ D x , Y 6 n +4 ] = [ D x , [ Y 1 , Y 6 n +3 ]] =  Y 1 , α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6 n +2  − −  Y 6 n +3 ,  Ae αu 0 + 2 B e − 2 αu 0  Y 2  = = α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6 n +3 +  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +4 . Therefore, w e hav e L 6 n +4 = L 6 n +3 ⊕ { Y 6 n +3 } , δ (6 n + 1) = 1. W e consider the op erators of the length 6 n + 4: Y 6( n +1) − 1 = [ Y 1 , Y 6 n +4 ] , ¯ Y 6( n +1) − 1 = [ Y 2 , Y 6 n +4 ] ,  Y 1 , ¯ Y 6 n +4  ,  Y 2 , ¯ Y 6 n +4  . Let us calculate the fo rm ulas by which the op erator D x comm utes with these op erators:  D x ,  Y 2 , ¯ Y 6 n +4  =  Y 2 , α 2  − Ae αu 0 − 4 B e − 2 αu 0  Y 6 n +2  = = α 3  − Ae αu 0 + 8 B e − 2 αu 0  Y 6 n +2 + α 2  − Ae αu 0 + 8 B e − 2 αu 0  [ Y 2 , Y 6 n +2 ] = = α 3  − Ae αu 0 + 8 B e − 2 αu 0  Y 6 n +2 =  D x , 2 α 2 Y 6 n +3 − α ¯ Y 6 n +4  . Th us w e hav e:  D x ,  Y 2 , ¯ Y 6 n +4  =  D x , 2 α 2 Y 6 n +3 − α ¯ Y 6 n +4  . Due to Lemma 1 w e obtain, that  Y 2 , ¯ Y 6 n +4  = 2 α 2 Y 6 n +3 − α ¯ Y 6 n +4 . F or ¯ Y 6 n +4 the follow ing form ula is satisfied:  D x ,  Y 1 , ¯ Y 6 n +4  = =  Y 1 , α 2  − Ae αu 0 − 4 B e − 2 αu 0  Y 6 n +2  − −  ¯ Y 6 n +4 ,  Ae αu 0 + B e − 2 αu 0  Y 2  = = α 2  − Ae αu 0 − 4 B e − 2 αu 0  Y 6 n +3 +  Ae αu 0 + B e − 2 αu 0   Y 2 , ¯ Y 6 n +4  = = α 2  − Ae αu 0 − 4 B e − 2 αu 0  Y 6 n +3 +  Ae αu 0 + B e − 2 αu 0   2 α 2 X 6 n +3 − α ¯ Y 6 n +4  = = α 2  Ae αu 0 − 2 B e − 2 αu 0  Y 6 n +3 − α  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +4 = [ D x , − αY 6 n +4 ] . Inte gr a b ility c onditions for two-dim ensional lattic es 20 Due to Lemma 1 we ha v e  Y 1 , ¯ Y 6 n +4  = − αY 6 n +4 . F or ¯ Y 6( n +1) − 1 the follow ing form ula is true:  D x , ¯ Y 6( n +1) − 1  = [ D x , [ Y 2 , Y 6 n +4 ]] = =  Y 2 , α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6 n +3 +  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +4  = = α 2  − Ae αu 0 − 4 B e − 2 αu 0  Y 6 n +3 + α  − Ae αu 0 + 2 B e − 2 αu 0  ¯ Y 6 n +4 + + α  Ae αu 0 − 2 B e − 2 αu 0  ¯ Y 6 n +4 +  Ae αu 0 + B e − 2 αu 0   Y 2 , ¯ Y 6 n +4  = = α 2  Ae αu 0 − 2 B e − 2 αu 0  Y 6 n +3 + α  − Ae αu 0 − B e − 2 αu 0  ¯ Y 6 n +4 = [ D x , − αY 6 n +4 ] . Due to Lemma 1, we conclude that ¯ Y 6( n +1) − 1 = − αY 6 n +4 . F or Y 6( n +1) − 1 the follow ing form ula is true:  D x , Y 6( n +1) − 1  = [ D x , [ Y 1 , Y 6 n +4 ]] = =  Y 1 , α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6 n +3 +  Ae αu 0 + B e − 2 αu 0  ¯ Y 6 n +4  − −  Y 6 n +4 ,  Ae αu 0 + B e − 2 αu 0  Y 2  = = α  − Ae αu 0 + 2 B e − 2 αu 0  Y 6 n +4 +  Ae αu 0 + B e − 2 αu 0   Y 1 , ¯ Y 6 n +4  + +  Ae αu 0 + B e − 2 αu 0  [ Y 2 , Y 6 n +4 ] = − 3 αAe αu 0 Y 6 n +4 . It is implies that the op erator Y 6( n +1) − 1 = Y 1 ... 121 is not linearly expressed through the op erators of lo w er order and L 6( n +1) − 1 = L 6 n +4 ⊕  Y 6( n +1) − 1  . T hus w e ha v e δ (6( n + 1 ) − 1) = 1 and Lemma 8 is prov ed. Eviden tly the Lemma 8 allo ws to complete the pro of o r t he first part of the Theorem 2. 5.2. The c ase α = − β No w w e pass to the second part o f the Theorem 2 . Here w e prov e that if p olynomial Λ( λ ) has t w o differen t nonzero ro ots α a nd β = − α then the Lie-Rinehart algebra L generated b y the op erator s X 0 and W 0 is of an infinite dimension. First w e intro duce t wo p o lynomials according to the rule Λ α ( λ ) = Λ( λ ) λ − α , Λ β ( λ ) = Λ( λ ) λ + α . Then w e construct t w o op erators P α , P β ∈ L : P α = Λ α ( ad X 0 W 0 ) , P β = Λ β ( ad X 0 W 0 ) and concen trate on the Lie-Rinehart a lg ebra L 1 ⊂ L b eing a subalgebra of L generated b y the op erators P α , P β . By construction these op erators satisfy the follow ing comm utativit y relations [ D x , P α ] = A ( u 1 , u − 1 ) e αu 0 X 0 , [ D x , P β ] = B ( u 1 , u − 1 ) e − αu 0 X 0 , where A = A ( u 1 , u − 1 ), B = B ( u 1 , u − 1 ) are some quasi-p olynomials in u 1 , u − 1 . W e assume that ( u − 1 , u 1 ) ∈ D , here D is a domain in C 2 , where b oth A , B do not v anish. Let us consider the op erators: Y 1 = P α + P β , Y 2 = ∂ ∂ u . Inte gr a b ility c onditions for two-dim ensional lattic es 21 F or these op erato rs the follo wing form ulas are true: [ D x , Y 1 ] =  Ae αu 0 + B e − αu 0  Y 2 , [ D x , Y 2 ] = 0 (5.20) Lemma 9. Assume that p olynomial Λ( λ ) defi ne d by (3.5) has two differ ent nonze r o r o ots α and − α . The n the fol lowing formulas hold: L 2 k + 1 = L 2 k ⊕ { Y 2 k + 1 } , L 2 k = L 2 k − 1 ⊕  Y 2 k , ¯ Y 2 k  . Pro of. Let us consider the op erator Y 3 = [ Y 2 , Y 1 ]. Using the Jacobi identit y and form ulas (5.2 0) w e prov e that [ D x , Y 3 ] = [ D x , [ Y 2 , Y 1 ]] =  Y 2 ,  Ae αu 0 + B e − αu 0  Y 2  = =  Aαe αu 0 − B αe − αu 0  Y 2 = α  Ae αu 0 − B e − αu 0  Y 2 . W e can see that Y 3 is not linearly expressed through the previous op erators. Th us L 3 = { Y 1 , Y 2 , Y 3 } . Let us construct the op erators of lenght 3: Y 4 = [ Y 1 , Y 3 ] = Y 121 , ¯ Y 4 = [ Y 2 , Y 3 ] = Y 221 . Using the Jak obi iden tit y and form ulas fo r t he op erators Y 1 , Y 2 , Y 3 w e find: [ D x , Y 4 ] = [ D x , [ Y 1 , Y 3 ]] =  Y 1 , α  Ae αu 0 − B e − αu 0  −  Y 3 ,  Ae αu 0 + B e − αu 0  Y 2  = = − α  Ae αu 0 − B e − αu 0  Y 3 +  Ae αu 0 + B e − αu 0  [ Y 2 , Y 3 ] = = − α  Ae αu 0 − B e − αu 0  Y 3 + α 2  Ae αu 0 + B e − αu 0  Y 1 ,  D x , ¯ Y 4  = [ D x , [ Y 2 , Y 3 ]] =  Y 2 , α  Ae αu 0 − B e − αu 0  Y 2  = = α 2  Ae αu 0 + B e − αu 0  Y 2 =  D x , α 2 Y 1  . Th us w e obta in the equality  D x , ¯ Y 4 − α 2 Y 1  = 0. Due to Lemma 1 w e conclude that ¯ Y 4 = α 2 Y 1 . The op erator Y 4 is not expressed through the op erators of lo w er o rder. So L 4 = { Y 1 , Y 2 , Y 3 , Y 4 } . No w w e construct the comm utators of length 4: Y 5 = [ Y 1 , Y 4 ] , ¯ Y 5 = [ Y 2 , Y 4 ] . No w w e need to calculate the f o rm ulas b y whic h the op erator D x comm utes with these op erators:  D x , ¯ X 5  = [ D x , [ X 2 , X 4 ]] =  X 2 , − α  Ae αu 0 − B e − αu 0  X 3 + α 2  Ae αu 0 + B e − αu 0  X 1  = − α  Ae αu 0 − B e − αu 0  ¯ X 4 + α 3  Ae αu 0 − B e − αu 0  X 1 = = − α  Ae αu 0 − B e − αu 0  α 2 X 1 + α 3  Ae αu 0 − B e − αu 0  X 1 = 0 . Based on Lemma 1 we claim that ¯ X 5 = 0 . W e hav e one more formula: [ D x , Y 5 ] = [ D x , [ Y 1 , Y 4 ]] ==  Y 1 , − α  Ae αu 0 − B e − αu 0  Y 3 + α 2  Ae αu 0 + B e − αu 0  Y 1  − −  Y 4 ,  Ae αu 0 + B e − αu 0  X 2  = = − α  Ae αu 0 − B e − αu 0  Y 4 +  Ae αu 0 + B e − αu 0  ¯ Y 5 = = − α  Ae αu 0 − B e − αu 0  Y 4 . Inte gr a b ility c onditions for two-dim ensional lattic es 22 Th us w e see that Y 5 is not linearly expressed thro ugh the op erators of lo w er order and L 5 = L 4 ⊕ { Y 5 } . No w w e construct the comm utators of length 5: Y 6 = [ Y 1 , Y 5 ] , ¯ Y 6 = [ Y 2 , Y 5 ] . These op erators a re satisfied the formulas: [ D x , Y 6 ] = [ D x , [ Y 1 , Y 5 ]] = =  Y 1 , − α  Ae αu 0 − B e − αu 0  Y 4  −  Y 5 ,  Ae αu 0 + B e − αu 0  Y 2  = = − α  Ae αu 0 − B e − αu 0  Y 5 +  Ae αu 0 + B e − αu 0  ¯ Y 6 . Th us we obta in t ha t L 6 = L 5 ⊕  Y 6 , ¯ Y 6  . Then w e consider the comm utators of length 6 : Y 7 = [ Y 1 , Y 6 ] , ¯ Y 7 = [ Y 2 , Y 6 ] ,  Y 1 , ¯ Y 6  ,  Y 2 , ¯ Y 6  The following formulas a r e true:  D x ,  Y 2 , ¯ Y 6  =  Y 2 , − α 2  Ae αu 0 + B e − αu 0  Y 4  = = − α 3  Ae αu 0 − B e − αu 0  Y 4 − α 2  Ae αu 0 + B e − αu 0  [ Y 2 , Y 4 ] = = − α 3  Ae αu 0 − B e − αu 0  Y 4 =  D x , α 2 Y 5  ,  D x ,  Y 1 , ¯ Y 6  =  Y 1 , − α 2  Ae αu 0 + B e − αu 0  Y 4  −  ¯ Y 6 ,  Ae αu 0 + B e − αu 0  X 2  = = − α 2  Ae αu 0 + B e − αu 0  Y 5 +  Ae αu 0 + B e − αu 0   Y 2 , ¯ Y 6  = = − α 2  Ae αu 0 + B e − αu 0  Y 5 +  Ae αu 0 + B e − αu 0  α 2 Y 5 = 0 ,  D x , ¯ Y 7  = [ D x , [ Y 2 , Y 6 ]] = =  Y 2 , − α  Ae αu 0 − B e − αu 0  Y 5 +  Ae αu 0 + B e − αu 0  ¯ Y 6  = = − α 2  Ae αu 0 + B e − αu 0  Y 5 +  Ae αu 0 + B e − αu 0   Y 2 , ¯ Y 6  = 0 , [ D x , Y 7 ] = [ D x , [ Y 1 , Y 6 ]] = =  Y 1 , − α  Ae αu 0 − B e − αu 0  Y 5 +  Ae αu 0 + B e − αu 0  ¯ Y 6  − −  Y 6 ,  Ae αu 0 + B e − αu 0  Y 2  = − α  Ae αu 0 − B e − αu 0  Y 6 + +  Ae αu 0 + B e − αu 0   Y 1 , ¯ Y 6  +  Ae αu 0 + B e − αu 0  [ Y 2 , Y 6 ] = = − α  Ae αu 0 − B e − αu 0  Y 6 . Th us w e see that ¯ Y 6 = α 2 Y 5 , ¯ Y 6 = 0, ¯ Y 7 = 0, the op era t or Y 7 is not linearly expressed through the o p erators of low er order and L 7 = L 6 ⊕ { Y 7 } , δ (7) = 1 . It can b e prov ed b y induction that [ Y 2 , Y i ] = 0, [ D x , Y i +1 ] = − α  Ae αu 0 − B e − αu 0  Y i , i = 3 , 4 , . . . , [ D x , [ Y 2 , Y i +1 ]] = − α 2  Ae αu 0 + B e − αu 0  Y i . Then w e hav e [ D x , [ Y 2 , [ Y 2 , Y i +1 ]]] =  Y 2 , − α 2  Ae αu 0 + B e − αu 0  Y 1  = = − α 3  Ae αu 0 − B e − αu 0  Y i = α 2 [ D x , Y i +1 ] . Inte gr a b ility c onditions for two-dim ensional lattic es 23 This equalit y implies [ D x , [ Y 2 , [ Y 2 , Y i +1 ]] − α 2 Y i +1 ] = 0. Using Lemma 1 we conclude that [ Y 2 , [ Y 2 , Y i +1 ]] = α 2 Y i +1 , i = 4 , 6 , . . . , 2 n . The following formula is true [ D x , [ Y 1 , [ Y 2 , Y i +1 ]]] =  Y 1 , − α 2  Ae αu 0 + B e − αu 0  Y i  − −  [ Y 2 , Y i +1 ] ,  Ae αu 0 + B e − αu 0  Y 2  = = − α 2  Ae αu 0 + B e − αu 0  Y i +1 +  Ae αu 0 + B e − αu 0  [ Y 2 , [ Y 2 , Y i +1 ]] = 0 . Due to Lemma 1 we obtain that [ Y 1 , [ Y 2 , Y i +1 ]] = 0, i = 4 , 6 , . . . 2 n . Th us we conclude that L 2 k + 1 = L 2 k ⊕ { Y 2 k + 1 } , L 2 k = L 2 k − 1 ⊕  Y 2 k , ¯ Y 2 k  . This completes the pro of of the Lemma 9. Now the second part of Theorem 2 immediately follo ws from the Lemma 9. 6. Conclusions In the ar t icle w e study the problem o f in tegrable classification of a r ather sp ecific but imp ortant class of tw o-dimensional lattices. W e used to this aim the metho d of Darb oux in tegrable reductions a nd the concept of characteristic Lie a lgebras [6]- [10]. By applying these implemen ts we deriv ed the necessary conditions of in tegrability for latt ices of t he form (1.1). 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