On the classification of Darboux integrable chains

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On the classification of Darboux integrable chains

Ismagil Habibullin,a兲 Natalya Zheltukhina,b兲and Aslı Pekcan

Department of Mathematics, Faculty of Science, Bilkent University, 06800 Ankara, Turkey

共Received 25 June 2008; accepted 10 September 2008; published online 9 October 2008兲

We study a differential-difference equation of the form tx共n+1兲= f共t共n兲,t共n+1兲,

tx共n兲兲 with unknown t=t共n,x兲 depending on x and n. The equation is called a

Darboux integrable if there exist functions F共called an x-integral兲 and I 共called an n-integral兲, both of a finite number of variables x,t共n兲,t共n⫾1兲,t共n⫾2兲, ... , tx共n兲,txx共n兲,..., such that DxF = 0 and DI = I, where Dx is the operator of total

differentiation with respect to x and D is the shift operator: Dp共n兲=p共n+1兲. The Darboux integrability property is reformulated in terms of characteristic Lie alge-bras that give an effective tool for classification of integrable equations. The com-plete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f共x,y,z兲=z+d共x,y兲. © 2008 American Institute of Physics. 关DOI:10.1063/1.2992950兴

I. INTRODUCTION

In this paper we study integrable semidiscrete chains of the following form:

tx共n + 1兲 = f共t共n兲,t共n + 1兲,tx共n兲兲, 共1兲

where the unknown t = t共n,x兲 is a function of two independent variables: discrete n and continuous x. Chain共1兲can also be interpreted as an infinite system of ordinary differential equations for the sequence of the variables兵t共n兲其_n=−⬁ _⬁. Here f = f共t,t1, tx兲 is assumed to be a locally analytical

func-tion of three variables satisfying at least locally the condifunc-tion

⳵f ⳵tx

⫽ 0. 共2兲

For the sake of convenience we introduce subindex denoting shifts tk= t共n+k,x兲共keep t0= t兲 and

derivatives tx=共⳵/⳵x兲t共n,x兲, txx=共⳵2/⳵x2兲t共n,x兲, and so on. We denote through D and Dxthe shift

operator and, correspondingly, the operator of total derivative with respect to x. For instance, Dh共n,x兲=h共n+1,x兲 and Dxh共n,x兲=共⳵/⳵x兲h共n,x兲. Set of all the variables 兵tk其k=−⬁ ⬁;兵Dx

m

t其_m=1⬁ con-stitutes the set of dynamical variables. Below we consider the dynamical variables as independent ones. Since in the literature the term “integrable” has various meanings let us specify the meaning used in the article. Introduce first notions of n- and x-integrals.1

Functions I and F, both depending on x and a finite number of dynamical variables, are called, respectively, n- and x-integrals of共1兲 if DI = I and DxF = 0.

Definition: Chain 共1兲 is called integrable 共Darboux integrable兲 if it admits a nontrivial n-integral and a nontrivial x-integral.

Darboux integrability implies the so-called C-integrability. Knowing both integrals F and I a Cole–Hopf-type differential substitution w = F + I reduces Eq. 共1兲 to the discrete version of the D’Alembert wave equation, w1x− wx= 0. Indeed,共D−1兲Dx共w兲=共D−1兲DxF + Dx共D−1兲I=0.

a兲_{Electronic mail: habibullin_i@mail.rb.ru. On leave from Ufa Institute of Mathematics, Russian Academy of Science,} Chernyshevskii Str., 112, Ufa 450077, Russia.

b兲_{Electronic mail: natalya@fen.bilkent.edu.tr.}

49, 102702-1

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It is remarkable that an integrable chain is reduced to a pair consisting of an ordinary differ-ential equation and an ordinary difference equation. To illustrate it, note first that any n-integral might depend only on x- and x-derivatives of the variable t, I = I共x,t,tx, txx, . . .兲, and similarly any

x-integral depends only on x and the shifts, F = F共x,t,t_⫾1, t_⫾2, . . .兲. Therefore each solution of integrable chain共1兲 satisfies two equations:

I共x,t,tx,txx, . . .兲 = p共x兲, F共x,t,t_⫾1,t_⫾2, . . .兲 = q共n兲,

with properly chosen functions p共x兲 and q共n兲.

Nowadays the discrete phenomena are studied intensively due to their various applications in physics. For the discussions and references we refer to the articles in Refs.1–5.

Chain 共1兲 is very close to a well studied object—the partial differential equation of the hyperbolic type

uxy= f共x,y,u,ux,uy兲. 共3兲

The definition of integrability for Eq. 共3兲 was introduced by Darboux. The famous Liouville equation uxy= euprovides an illustrative example of the Darboux integrable equation. An effective

criterion of integrability of共3兲 was discovered by Darboux himself: Eq.共3兲 is integrable if and only if the Laplace sequence of the linearized equation terminates at both ends共see Refs.6–8兲. This criterion of integrability was used in Ref.8where the complete list of all Darboux integrable equations of form共3兲is given.

An alternative approach to the classification problem based on the notion of the characteristic Lie algebra of hyperbolic-type systems was introduced years ago in Refs. 9 and 10. In these articles an algebraic criterion of Darboux integrability property has been formulated. An important classification result was obtained in Ref.9for the exponential system

uxy i _{= exp共a}

i1u1+ ai2u2+ ¯ + ainun兲, i = 1,2, ... ,n. 共4兲

It was proved that system 共4兲 is a Darboux integrable if and only if the matrix A =共aij兲 is the

Cartan matrix of a semisimple Lie algebra. Properties of the characteristic Lie algebras of the hyperbolic systems uxy i = cjk i ujuk, i, j,k = 1,2, . . . ,n, 共5兲

have been studied in Refs.11and12. Hyperbolic systems of general form admitting integrals are studied in Ref.13. A promising idea of adopting the characteristic Lie algebras to the problem of classification of the hyperbolic systems which are integrated by means of the inverse scattering transforms method is discussed in Ref.14.

The method of characteristic Lie algebras is closely connected with the symmetry approach15 which is proved to be a very effective tool to classify integrable nonlinear equations of evolution-ary type16–20 共see also the survey in Ref. 3 and references therein兲. However, the symmetry approach meets very serious difficulties when applied to hyperbolic-type models. After the papers in Refs.21and22it became clear that this case needs alternative methods.

In this article an algorithm of classification of integrable discrete chains of form 共1兲 is sug-gested based on the notion of the characteristic Lie algebra共see also Refs.23–25兲.

To introduce the characteristic Lie algebra Lnof 共1兲in the direction of n, note that

D−j ⳵

⳵t₁D

j_{I = 0} _共6兲

for any n-integral I and jⱖ1. Indeed, the equation DI=I can be rewritten in an enlarged form as I共x,n + 1,t1, f, fx, fxx, . . .兲 = I共x,n,t,tx,txx, . . .兲. 共7兲

The left hand side DI of equality共7兲contains the variable t₁, while the right hand side does not. Hence,共⳵/⳵t₁兲共DI兲=0, which implies

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D−1 ⳵ ⳵t1

DI = 0.

Proceeding this way one can easily prove共6兲from the equality Dj_{I = I, j}_ⱖ1.

Define vector fields

Yj= D−j ⳵ ⳵t1 Dj, jⱖ 1, 共8兲 and Xj= ⳵ ⳵t_−j, jⱖ 1. 共9兲

We have YjI = 0 and XjI = 0 for any n-integral I of共1兲and jⱖ1. The following theorem 共see Ref. 24兲 defines the characteristic Lie algebra Lnof 共1兲.

Theorem 1: Equation (1) admits a nontrivial n -integral if and only if the following two conditions hold:

共1兲 Linear space spanned by the operators 兵Yj其1⬁is of finite dimension. We denote this dimension

by N .

共2兲 Lie algebra Ln generated by the operators Y1, Y2, . . . , YN, X1, X2, . . . , XN is of finite

dimen-sion. We call Ln the characteristic Lie algebra of(1) in the direction of n .

To introduce the characteristic Lie algebra Lxof共1兲in the direction of x, note that Eq.共1兲due

to共2兲 can be rewritten as tx共n−1兲=g共t共n兲,t共n−1兲,tx共n兲兲. An x-integral F共x,t,t_⫾1, t_⫾2, . . .兲 solves

the equation DxF = 0, i.e., K0F = 0, where

K₀= ⳵ ⳵x+ tx ⳵ ⳵t+ f ⳵ ⳵t1 + g ⳵ ⳵t−1 + f₁ ⳵ ⳵t₂+ g−1 ⳵ ⳵t₋₂+ ¯ . 共10兲

Since F does not depend on the variable txone gets XF = 0, where

X = ⳵ ⳵tx

. 共11兲

Therefore, any vector field from the Lie algebra generated by K0and X annulates F. This algebra

is called the characteristic Lie algebra Lxof chain共1兲in the x-direction.

The following result is essential. Its proof is a simple consequence of the famous Jacobi theorem共the Jacobi theorem is discussed, for instance, in Ref.10兲.

Theorem 2: Equation(1)admits a nontrivial x -integral if and only if its Lie algebra Lxis of

finite dimension.

In the present paper we restrict ourselves to consideration of the existence of x-integrals for a particular kind of chain共1兲, namely, we study chains of the form

t1x= tx+ d共t,t1兲共12兲

admitting nontrivial x-integrals. The main result of the paper, Theorem 3 below, is the complete list of chains共12兲admitting nontrivial x-integrals.

Theorem 3: Chain (12) admits a nontrivial x -integral if and only if d共t,t1兲 is one of the

following kinds: 共1兲 d共t,t1兲=A共t−t1兲 ,

共2兲 d共t,t1兲=c0t共t−t1兲+c2共t−t1兲2+ c3t − c3t1,

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共4兲 d共t,t1兲=c4共e␣t1− e␣t兲+c5共e−␣t1− e−␣t兲 ,

where A = A共t−t1兲 is a function of␶= t − t1and c0- c5are some constants with c0⫽0, c4⫽0,

and c5⫽0, and␣ is a nonzero constant. Moreover, x-integrals in each of the cases are

共i兲 F = x +兰␶du/A共u兲 if A共u兲⫽0 and F=t1− t if A共u兲⬅0 ,

共ii兲 F =共1/共−c2− c0兲兲ln兩共−c2− c0兲␶1/␶2+ c2兩+1/c2ln兩c2␶1/␶− c2− c0兩 for c2共c2+ c0兲⫽0 , F=ln␶1

− ln␶2+␶1/␶for c2= 0 , and F =␶1/␶2− ln␶+ ln␶1for c2= −c0 ,

共iii兲 F =兰␶e−␣udu/A共u兲−兰␶1_du/A共u兲 , and

共iv兲 F =关共e␣t− e␣t2兲共e␣t1_{− e}␣t3兲兴/关共e␣t_{− e}␣t3兲共e␣t1_{− e}␣t2兲兴 .

The n-integrals of chain 共12兲can be studied in a similar way by using Theorem 1, but this problem is out of the frame of the present article.

The article is organized as follows. In Sec. II, by using the properly chosen sequence of multiple commutators, a very rough classification result is obtained: function d共t,t1兲 for chain共12兲

admitting x-integrals is a quasipolynomial on t with coefficients depending on␶= t − t1. Then it is

observed that the exponents␣0= 0 ,␣1, . . . ,␣sin expansion共26兲cannot be arbitrary. For example,

if the coefficient before e␣0t= 1 is not identically zero, then the quasipolynomial d共t,t

1兲 is really a

polynomial on t with coefficients depending on ␶. In Sec. III we prove that the degree of this polynomial is at most 1. If d contains a term of the form␮共␶兲tje␣kt_with␣

k⫽0, then j=0 共Sec. IV兲.

In Sec. V it is proved that if d contains terms with e␣kt _{and e}␣jt _{having nonzero exponents, then}

␣k= −␣j. This last case contains chains having infinite dimensional characteristic Lie algebras for

which the sequence of multiple commutators grows very slowly. They are studied in Secs. VI and VII. One can find the well known semidiscrete version of the sine-Gordon 共SG兲 model among them. It is worth mentioning that in Sec. VII the characteristic Lie algebra Lxfor semidiscrete SG

is completely described. The last section, Sec. VIII, contains the proof of the main theorem, Theorem 3, and here the method of constructing x-integrals is also briefly discussed.

II. A NECESSARY INTEGRABILITY CONDITION

Define a class F of locally analytical functions each of which depends only on a finite number of dynamical variables. In particular, we assume that f共t,t1, tx兲苸F. We will consider vector fields

given as an infinite formal series of the form

Y =

兺

−⬁ ⬁ yk ⳵ ⳵tk , 共13兲

with coefficients yk苸F. Introduce notions of linearly dependent and independent sets of vector

fields共13兲. Denote through PNthe projection operator acting according to the rule

PN共Y兲 =

兺

k=−N N yk ⳵ ⳵tk . 共14兲

First we consider finite vector fields as

Z =

兺

k=−N N zk ⳵ ⳵tk . 共15兲

We say that a set of finite vector fields Z1, Z2, . . . , Zmis linearly dependent in some open region U

if there is a set of functions ␭1,␭2, . . . ,␭mdefined on U such that the function 兩␭1兩2+兩␭2兩2+¯

+兩␭m兩2does not vanish identically and the condition

␭1Z1+␭2Z2+ ¯ + ␭mZm= 0 共16兲

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We call a set of the vector fields Y1, Y2, . . . , Ymof form共13兲linearly dependent in region U if

for each natural N the set of finite vector fields PN共Y1兲, PN共Y2兲, ... , PN共Ym兲 is linearly dependent

in this region. Otherwise we call the set Y1, Y2, . . . , Ymlinearly independent in U.

The following proposition is very useful. Its proof is almost evident. Proposition: If a vector field Y is expressed as a linear combination,

Y =␭₁Y₁+␭₂Y₂+ ¯ + ␭mYm, 共17兲

where the set of vector fields Y1, Y2, . . . , Ymis linearly independent in U and the coefficients of all

the vector fields Y , Y1, Y2, . . . , Ym belonging to F are defined in U , then the coefficients

␭1,␭2, . . . ,␭mare in F .

Below we concentrate on the class of chains of form 共12兲. For this case the Lie algebra Lx

splits down into a direct sum of two subalgebras. Indeed, since f = tx+ d and g = tx− d−1 one gets

fk= tx+ d +兺j=1 k

dj and g−k= tx−兺j=1 k+1

d−k for kⱖ1, where d=d共t,t1兲 and dj= d共tj, tj+1兲. Due to this

observation the vector field K0 can be rewritten as K0= txX˜ +Y, with

X ˜ = ⳵ ⳵t+ ⳵ ⳵t₁+ ⳵ ⳵t₋₁+ ⳵ ⳵t₂+ ⳵ ⳵t₋₂+¯ 共18兲 and Y = ⳵ ⳵x+ d ⳵ ⳵t₁− d−1 ⳵ ⳵t₋₁+共d + d1兲 ⳵ ⳵t₂−共d−1+ d−2兲 ⳵ ⳵t₋₂+ ¯ .

Due to the relations 关X,X˜兴=0 and 关X,Y兴=0 we have X˜=关X,K0兴苸Lx; hence Y苸Lx. Therefore

Lx=兵X其丣Lx1, where Lx1is the Lie algebra generated by the operators X˜ and Y.

Lemma 1: If Eq.(12)admits a nontrivial x -integral, then it admits a nontrivial x -integral F such that⳵F/⳵x = 0 .

Proof: Assume that a nontrivial x-integral of共12兲exists. Then the Lie algebra L_x1is of finite dimension. One can choose a basis of L_x1in the form

re-duces the system to the form

⳵F ⳵x +

兺

_k=0 N−1 a ˜_1,k⳵F ⳵␪k = 0,

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⳵F ⳵␪k = 0, 2ⱕ j ⱕ N − 2, which is equivalent to ⳵F ⳵x + a˜1,N−1 ⳵F ⳵␪N−1 = 0 for F = F共x,␪N−1兲.

There are two possibilities: 共1兲 a˜1,N−1= 0 and 共2兲 a˜1,N−1⫽0. In case 共1兲, we at once have

⳵F/⳵x = 0. In case 共2兲, F=x+H共␪N−1兲=x+H共t,t1, . . . , tN−1兲 for some function H. Evidently, F1

= DF = x + H共t1, t2, . . . , tN兲 is also an x-integral, and F1− F is a nontrivial x-integral not depending

on x. 䊐

Below we look for x-integrals F depending on dynamical variables t , t_⫾1, t_⫾2, . . . only共not depending on x兲. In other words, we study the Lie algebra generated by vector fields X˜ and Y˜, where Y ˜ = d ⳵ ⳵t₁− d−1 ⳵ ⳵t₋₁+共d + d1兲 ⳵ ⳵t₂−共d−1+ d−2兲 ⳵ ⳵t₋₂+ ¯ . 共19兲

One can prove that the linear operator Z→DZD−1 _{defines an automorphism of the characteristic}

Lie algebra Lx. This automorphism plays the crucial role in all of our further considerations.

Further we refer to it as the shift automorphism. For instance, direct calculations show that

DX˜ D−1= X˜ , DY˜D−1= − dX˜ + Y˜ . 共20兲

Lemma 2: Suppose that a vector field of the form Z =兺a共j兲共⳵/⳵tj兲 with the coefficients a共j兲

= a共j,t,t⫾1, t⫾2, . . .兲 depending on a finite number of the dynamical variables solves an equation of

the form DZD−1=␭Z . If for some j= j0we have a共j0兲⬅0 , then Z=0 .

Proof: By applying the shift automorphism to the vector field Z one gets DZD−1

=兺D共a共j兲兲共⳵/⳵tj+1兲. Now, to complete the proof, we compare the coefficients of ⳵/⳵tj in the

equation兺D共a共j兲兲共⳵/⳵tj+1兲=␭兺a共j兲共⳵/⳵tj兲. 䊐

Construct an infinite sequence of multiple commutators of the vector fields X˜ and Y˜, Y

˜

1=关X˜,Y˜兴, Y˜k=关X˜,Y˜k−1兴 for k ⱖ 2. 共21兲 Lemma 3: We have

DY˜kD−1= − X˜k共d兲X˜ + Y˜k, kⱖ 1. 共22兲 Proof: We prove the statement by induction on k. The base of induction holds. Indeed, by共20兲 and共21兲, we have

DY˜1D−1= D关X˜,Y˜兴D−1=关DX˜D−1,DY˜ D−1兴 = 关X˜,− dX˜ + Y˜兴 = − X˜共d兲X˜ + Y˜1.

Assuming Eq.共22兲holds for k = n − 1, we have

DYñD−1=关DX˜D−1,DYñ−1D−1兴 = 关X˜,− Xñ−1共d兲X˜ + Yñ−1兴 = − Xñ共d兲X˜ + Yñ,

which finishes the proof of the lemma. 䊐

Since vector fields X, X˜ , and Y˜ are linearly independent, then the dimension of Lie algebra Lx

is at least 3. By 共22兲, case Y˜1= 0 corresponds to X˜ 共d兲=0, or dt+ dt₁= 0, which implies d = A共t

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Assume Eq.共12兲admits a nontrivial x-integral and Y˜₁⫽0. Consider the sequence of the vector fields 兵Y˜₁, Y˜₂, Y˜₃, . . .其. Since Lxis of finite dimension, then there exists a natural number N such

that

Y ˜

N+1=␥1Y˜1+␥2Y˜2+ ¯ +␥NY˜N, Nⱖ 1, 共23兲

and Y˜₁, Y˜₂, . . . , Y˜Nare linearly independent. Therefore,

DY˜N+1D−1= D共␥1兲DY˜1D−1+ D共␥2兲DY˜2D−1+ ¯ + D共␥N兲DY˜ND−1, Nⱖ 1.

Due to Lemma 3 and共23兲the last equation can be rewritten as

− X˜N+1共d兲X˜ +␥1Y˜1+␥2Y˜2+ ¯ +␥NY˜N= D共␥1兲共− X˜共d兲X˜ + Y˜1兲 + D共␥2兲共− X˜2共d兲X˜ + Y˜2兲

+ ¯ + D共␥N兲共− X˜N共d兲X˜ + Y˜N兲.

Comparing coefficients before linearly independent vector fields X˜ ,Y˜1, Y˜2, . . . , Y˜N, we obtain the

following system of equations: X

˜N+1_{共d兲 = D共}_␥

1兲X˜共d兲 + D共␥2兲X˜2共d兲 + ¯ + D共␥N兲X˜N共d兲,

␥1= D共␥1兲, ␥2= D共␥2兲, ... , ␥N= D共␥N兲.

Since the coefficients of the vector fields Y˜jdepend only on the variables t , t⫾1, t⫾2, . . . the factors

␥j might depend only on these variables共see the proposition above兲. Hence the system of

equa-tions implies that all coefficients ␥k, 1ⱕkⱕN, are constants, and d=d共t,t1兲 is a function that

satisfies the following differential equation: X

˜N+1_{共d兲 =}_␥

1X˜ 共d兲 +␥2X˜2共d兲 + ¯ +␥NX˜N共d兲, 共24兲

where X˜ 共d兲=dt+ dt₁. Using the substitution s = t and␶= t − t1, Eq.共24兲can be rewritten as

⳵N+1_d ⳵sN+1=␥1 ⳵d ⳵s+␥2 ⳵2_d ⳵s2+ ¯ +␥N ⳵N d ⳵sN, 共25兲

which implies that

d共t,t1兲 =

兺

k

冉

兺

j=0 m_k−1

␭k,j共t − t1兲tj

冊

e␣kt 共26兲

for some functions␭k,j共t−t1兲, where␣kare roots of multiplicity mkfor a characteristic equation of 共25兲.

Let␣0= 0 ,␣1, . . . ,␣sbe the distinct roots of characteristic equation共24兲. Equation共24兲can be

rewritten as

⌳共X˜兲d ª X˜m₀_{共X˜ −}_␣

1兲m1共X˜ −␣2兲m2¯ 共X˜ −␣s兲msd = 0 共27兲

and m0+ m1+¯ +ms= N + 1, m0ⱖ1.

Initiated by formula 共19兲 define a map h→Yh which assigns to any function h

= h共t,t_⫾1, t_⫾2, . . .兲 a vector field Yh= h ⳵ ⳵t1 − h−1 ⳵ ⳵t−1 +共h + h1兲 ⳵ ⳵t2 −共h−1+ h−2兲 ⳵ ⳵t−2 + ¯ .

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P共adX˜兲Y˜ = YP共X˜兲h, where adXY =关X,Y兴, 共28兲

which establishes an isomorphism between the linear space V of all solutions of Eq.共25兲and the linear space V˜ =span兵Y˜ ,Y˜₁, . . . , Y˜N其 of the corresponding vector fields.

Represent function 共26兲 as the sum d共t,t₁兲= P共t,t₁兲+Q共t,t₁兲 of the polynomial part P共t,t₁兲 =兺m_j=00−1␭

0,j共t−t1兲tj and the “exponential” part Q共t,t1兲=兺k=1 s _共兺

j=0 m_k−1_␭

k,j共t−t1兲tj兲e␣kt.

Lemma 4: Assume that Eq. (12)admits a nontrivial x -integral. Then one of the functions P共t,t₁兲 and Q共t,t₁兲 vanishes.

Proof: Assume in contrary that neither of the functions vanish. First we show that in this case algebra Lx contains vector fields T0= YA共␶兲e␣kt and T1= YB共␶兲 for some functions A共␶兲 and B共␶兲.

Indeed, take T0ª⌳0共adX˜兲Y˜ =Y_⌳₀_共X˜兲d苸Lx, where ⌳0共␭兲=⌳共␭兲/共␭−␣k兲. Evidently the function

A

˜ 共t,t1兲=⌳0共X˜兲d solves the equation 共X˜−␣k兲A˜共t,t1兲=⌳共X˜兲d=0 which implies immediately that

A

˜ 共t,t1兲=A共␶兲e␣kt. In a similar way one shows that T1苸Lx. Note that due to our assumption the

functions A共␶兲 and B共␶兲 cannot vanish identically.

Consider an infinite sequence of the vector fields defined as follows: T2=关T0,T1兴, T3=关T0,T2兴, ... , Tn=关T0,Tn−1兴, n ⱖ 3.

One can show that

关X˜,T0兴 =␣kT0, 关X˜,T1兴 = 0, 关X˜,Tn兴 =␣k共n − 1兲Tn, nⱖ 2, DT0D−1= − Ae␣ktX˜ + T0, DT1D−1= − BX˜ + T1, DTnD−1= Tn− 共n − 1兲共n − 2兲 2 ␣kAe ␣kt_T n−1+ bnX˜ +

兺

k=0 n−2 ak共n兲Tk, nⱖ 2.

Since algebra Lxis of finite dimension, then there exists number N such that

TN+1=␭X˜ +␮0T0+␮1T1+ ¯ +␮NTN, 共29兲

and vector fields X˜ ,T0, T1, . . . , TNare linearly independent. We have

DTN+1D−1= D共␭兲X˜ + D共␮0兲兵− Ae␣ktX˜ + T0其 + ¯ + D共␮N兲

再

TN− 共N − 1兲共N − 2兲 2 ␣kAe ␣kt_T N−1+¯

冎

.

By comparing the coefficients before TN in the last equation one gets

␮N−

N共N − 1兲 2 ␣kA共␶兲e

␣kt= D共␮

N兲.

It follows that␮Nis a function of variable t only. Also, by applying adX˜ to both sides of Eq.共29兲,

one gets

N␣kTN+1=关X˜,TN+1兴 = X˜共␭兲X˜ + 共X˜共␮0兲 +␮0␣k兲T0+ ¯ + 共X˜ 共␮N兲 +␮N共N − 1兲␣k兲TN.

Again, by comparing coefficients before TN, we have

N␣k␮N= X˜ 共␮N兲 + 共N − 1兲␣k␮N, i . e . , X˜ 共␮N兲 =␣k␮N.

Therefore, ␮N= A1e␣kt, where A1 is some nonzero constant, and thus A共␶兲e␣kt= A2e␣kt− A2e␣kt1.

Here A₂ is some constant. We have T₀= A₂e␣kt_X˜ −A

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S0=

兺

j=−⬁ ⬁ e␣ktj ⳵ ⳵tj . Also, 关X˜,S0兴 =␣kS0, DS0D−1= S0.

Consider a new sequence of vector fields,

P1= S0, P2=关T1,S0兴, P3=关T1, P2兴, Pn=关T1, Pn−1兴, n ⱖ 3.

One can show that

关X˜,Pn兴 =␣kPn, DPnD−1= Pn−␣k共n − 1兲BPn−1+ bnX˜ + anS0+

兺

j=2 n−2

aj共n兲Pj, nⱖ 2.

Since algebra Lxis of finite dimension, then there exists number M such that

P_M+1=␭ⴱX˜ +␮₂ⴱP₂+ ¯ +␮MⴱPM, 共30兲

and fields X˜ , P₂, . . . , PM are linearly independent. Thus,

DP_M+1D−1_{= D}_共␭ⴱ_{兲X˜ + D共}_␮ 2 ⴱ_兲兵P

2+¯其 + ¯ + D共␮ⴱM兲兵PM−␣k共M − 1兲BPM−1+¯其.

We compare the coefficients before PM in the last equation and get

␮ⴱM− M␣kB共␶兲 = D共␮Mⴱ兲, 共31兲

which implies that␮Mⴱ is a function of variable t only. Also, by applying adX˜ to both sides of共30兲,

one gets

␣kPM+1=关X˜,PM+1兴 = X˜共␭ⴱ兲X˜ + 共X˜共␮2ⴱ兲 +␣k␮ⴱ2兲P2+ ¯ + 共X˜ 共␮Mⴱ兲 +␣k␮Mⴱ兲PM.

Again, we compare the coefficients before PM and have ␣k␮Mⴱ共t兲=X˜共␮Mⴱ共t兲兲+␣k␮Mⴱ共t兲, which

implies that␮Mⴱ is a constant. It follows then from共31兲that B共␶兲=0. This contradiction shows that

our assumption that both functions are not identically zero was wrong. 䊐 III. MULTIPLE ZERO ROOT

In this section we assume that Eq.共12兲admits a nontrivial x-integral and that␣0= 0 is a root

of the characteristic polynomial⌳共␭兲. Then, due to Lemma 4, zero is the only root and therefore ⌳共␭兲=␭m+1_{. It follows from formula}_共26兲_{with m}

0= m + 1 that

d共t,t₁兲 = a共␶兲tm_{+ b}_共_␶_兲tm−1₊ _{¯ , m = m}

0− 1ⱖ 0.

The case m = 0 corresponds to a very simple equation, t1x= tx+ A共t−t1兲, which is easily solved in

quadratures, so we concentrate on the case mⱖ1. For this case the characteristic algebra Lx

contains a vector field T = Y_␬˜with

␬ ˜ = a共␶兲t + 1 mb共␶兲. Indeed, T = 1 m!adX˜ m−1 Y ˜ = Y_␬˜. 共32兲

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T0= X˜ , T1=关T,T0兴 = Y−a共␶兲, Tk+1=关T,Tk兴, k ⱖ 0, Tk,0=关T0,Tk兴.

Note that T_1,0= 0. We will see below that the linear space spanned by this sequence is not invariant under the action of the shift automorphism Z→DZD−1_{introduced above. We extend the sequence}

to provide the invariance property. We define T_␣ with the multi-index ␣. For any sequence ␣ = k , 0 , i1, i2, . . . , in−1, in, where k is any natural number, ij苸兵0;1其, denote

T_␣=

再

关T0 ,T_k,0,i 1,. . .,in−1兴 if in= 0 关T,Tk,0,i₁,. . .,i_n−1兴 if in= 1,

冎

m共␣兲 =

冦

k if ␣= k k if ␣= k,0 k + i₁+ . . . + in if ␣= k,0,i1, . . . ,in,

冧

l共␣兲 = k + n + 1 − m共␣兲.

The multi-index␣is characterized by two quantities, m共␣兲 and l共␣兲, which allow to order partially the sequence兵T_␣其. We have

DT0D−1= T0, DTD−1= T −˜ T␬ 0, DT1D−1= T1+ aT0.

One can prove by induction on k that

DTkD−1= Tk+ aTk−1−␬˜

兺

m共␤兲=k−1

T_␤+

兺

m共␤兲ⱕk−2

␩共k,␤兲T␤. 共33兲

In general, for any␣,

DT_␣D−1= T_␣+

兺

m共␤兲ⱕm共␣兲−1

␩共␣,␤兲T_␤. 共34兲

We can choose a system P of linearly independent vector fields in the following way: 共1兲 T and T0are linearly independent. We take them into P.

共2兲 We check whether T, T0, and T1are linearly independent or not. If they are dependent, then

P =兵T,T0其 and T1=␮T +␭T0for some functions␮and␭.

共3兲 If T, T0, and T1are linearly independent, then we check whether T, T0, and T1, T2are linearly

independent or not. If they are dependent, then P =兵T,T0, T1其.

共4兲 If T, T0, T1, and T2 are linearly independent, we add vector fields T␤, m共␤兲=2, ␤苸I2

共actually, by definition I2 is the collection of such ␤兲, in such a way that J2

ª兵T,T0, T1, T2,艛␤苸I₂T␤其 is a system of linearly independent vector fields and for any T␥

with m共␥兲ⱕ2 we have T␥=兺T_␤苸J₂␮共␥,␤兲T␤.

共5兲 We check whether T3艛J2is a linearly independent system. If it is not, then P consists of all

elements from J2, and T3=兺T_␤苸J₂␮共␥,␤兲T␤. If it is, then to the system T3艛J2we add vector

fields T_␤, m共␤兲=3,␤苸I3, in such a way that J3ª兵T3, J2,艛␤苸I₃T␤其 is a system of linearly

independent vector fields and for any T_␥with m共␥兲ⱕ3 we have T_␥=兺T_␤苸J₃␮共␥,␤兲T␤.

We continue the construction of system P. Since Lxis of finite dimension, then there exists

such a natural number N such that we have the following: 共i兲 Tk苸 P, kⱕN.

共ii兲 m共␤兲ⱕN for any T_␤苸 P.

共iii兲 For any T_␥with m共␥兲ⱕN we have T_␥=兺T_␤苸P,m共␤兲ⱕm共␥兲␮共␥,␤兲T␤ and also

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We then have the following:

共iv兲 For any vector field T_␣ with m共␣兲=N that does not belong to P, the coefficient ␮共␣, N兲 before TNin the expansion

T_␣=␮共␣,N兲TN+

兺

T_␤苸P ␮共␣,␤兲T_␤ 共35兲 is constant. Indeed, by共34兲, DT_␣D−1_{= T} ␣+

兺

m共␤兲ⱕN−1 ␩共␣,␤兲T_␤=␮共␣,N兲TN+

兺

T_␤苸P ␮共␣,␤兲T_␤+

兺

m共␤兲ⱕN−1 ␩共␣,␤兲T_␤. From共35兲we also have

DT_␣D−1= D共␮共␣,N兲兲DTND−1+

兺

T_␤苸P D共␮共␣,␤兲兲DT_␤D−1= D共␮共␣,N兲兲兵TN+¯其 +

兺

T_␤_苸P D共␮共␣,␤兲兲兵T_␤+¯其.

By comparing the coefficients before TN in these two expressions for DT␣D−1, we have

␮共␣,N兲 = D共␮共␣,N兲兲, which implies that␮共␣, N兲 is a constant indeed.

Lemma 5: We have a共␶兲=c₀␶+ c₁, where c₀ and c₁ are some constants.

Proof: Since T_N+1=␮共N + 1,N兲TN+

兺

T_␤苸P ␮共N + 1,␤兲T␤, then DTN+1D−1= D共␮共N + 1,N兲兲兵TN+¯其 +

兺

T_␤_苸P D共␮共N + 1,␤兲兲兵T_␤+¯其. On the other hand,

a共␶兲t + 1

mb共␶兲

冊

=␮N共t1,n + 1兲 −␮N共t,n兲.

By differentiating both sides of the equation with respect to t and then t1, we have

− a

⬙

共␶兲 − c

冉

Lemma 6: If the Lie algebra generated by the vector fields S0=兺⬁j=−⬁⳵/⳵wj and P

=兺⬁_j=−_⬁c共wj兲共⳵/⳵wj兲 is of finite dimension, then c共w兲 is one of the following forms:

共1兲 c共w兲=c2+ c3e␭w+ c4e−␭w,␭⫽0 , and

共2兲 c共w兲=c2+ c3w + c4w2, where c2− c4are some constants.

Proof: Introduce vector fields

S₁=关S₀, P兴, S₂=关S₀,S₁兴, ... , Sn=关S0,Sn−1兴, n ⱖ 3. Clearly, we have Sn=

兺

j=−⬁ ⬁ c共n兲共wj兲 ⳵ ⳵wj , nⱖ 1. 共39兲

Since all vector fields Snare elements of Lxand Lxis of finite dimension, then there exists a natural

number N such that

SN+1=␮NSN+␮N−1SN−1+ ¯ +␮1S1+␮0P +␮S0, 共40兲

and S₀, P , S₁, . . . , SNare linearly independent.共Note that we may assume that S0and P are linearly

independent兲. Since DS0D−1= S0, DPD−1= P, and DSnD−1= Snfor any nⱖ1, then it follows from 共40兲that

SN+1= D共␮N兲SN+ D共␮N−1兲SN−1+ ¯ + D共␮1兲S1+ D共␮0兲P + D共␮兲S0

and together with共40兲, it implies that␮,␮0,␮1, . . . ,␮Nare all constants.

By comparing the coefficients before⳵/⳵w in共40兲one gets, with the help of共39兲, the follow-ing equality:

c共N+1兲共w兲 =␮Nc共N兲共w兲 + ¯ +␮1c

⬘

共w兲 +␮0c共w兲 +␮.

Thus, c共w兲 is a solution of the nonhomogeneous linear differential equation with constant coeffi-cient whose characteristic polynomial is

⌳共␭兲 = ␭N+1₋_␮

N␭N− ¯ −␮1␭ −␮0.

Denote by␤₁,␤₂, . . . ,␤tthe characteristic roots and by m1, m2, . . . , mttheir multiplicities.

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共i兲 There exists a nonzero characteristic root, say,␤1, and its multiplicity m1ⱖ2.

共ii兲 There exists zero characteristic root, say,␤₁, and m₁ⱖ3,␮= 0 or m₁ⱖ2,␮⫽0. 共iii兲 There are two distinct characteristic roots, say,␤₁and␤₂ with␤₁⫽0 and␤₂= 0. 共iv兲 There are two nonzero distinct characteristic roots, say,␤1 and␤2.

In case 共i兲, consider

⌳1共␭兲 = ⌳共␭兲 ␭ −␤1 and ⌳₁共2兲共␭兲 = ⌳共␭兲共␭ −␤1兲2 . Then⌳1共S0兲c共w兲=␣1e␤1w+␣2and⌳1 共2兲_共S

0兲c共w兲=共␣3w +␣4兲e␤1w+␣5, where␣j, 1ⱕ jⱕ5, are some

+␣4P1+␣5S0 =␣3P2+␣4P1+␣5S0

are elements from Lx and therefore vector fields P1=兺j=−⬁ ⬁e␤1 w_j_共_⳵_/_⳵_w

j兲 and P2

=兺⬁_j=−_⬁wje␤1wj共⳵/⳵wj兲 belong to Lx. Since P1and P2generate an infinite dimensional Lie algebra

Lx, then case 共i兲 fails to be true.

In case 共ii兲, consider ⌳1共3兲共␭兲 = ⌳共␭兲 ␭3 and ⌳1共2兲共␭兲 = ⌳共␭兲 ␭2 if ␮= 0 or ⌳1共3兲共␭兲 = ⌳共␭兲 ␭2 and ⌳1共2兲共␭兲 = ⌳共␭兲 ␭ if ␮⫽ 0. We have ⌳1共3兲共S0兲c共w兲 =␣1w3+␣2w2+␣3w +␣4 and ⌳1共2兲共S0兲c共w兲 =␣5w2+␣6w +␣7,

where␣j, 1ⱕ jⱕ7, are some constants with␣1⫽0 and␣5⫽0. Straightforward calculations show

that vector fields

⌳1共3兲共adS₀兲P =

兺

j=−⬁ ⬁ 共␣1w3j+␣2wj2+␣3wj+␣4兲 ⳵ ⳵wj and ⌳₁共2兲共adS₀兲P =

兺

j=−⬁ ⬁ 共␣5wj 2 +␣6wj+␣7兲 ⳵ ⳵wj

generate an infinite dimensional Lie algebra. It proves that case共ii兲 fails to be true. In case 共iii兲, consider

⌳1共␭兲 = ⌳共␭兲 ␭ −␤1 and ⌳2共␭兲 = ⌳共␭兲 ␭ . We have

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⌳1c共w兲 =␣1e␤1w+␣2 and ⌳2c共w兲 =␣3w +␣4 if ␮= 0

or

⌳1共S0兲c共w兲 =␣1e␤1

w

+␣₂ and ⌳₂共S₀兲c共w兲 =␣₅w2+␣₆w +␣₇ if ␮⫽ 0,

where␣j, 1ⱕ jⱕ7, are constants with ␣1⫽0,␣3⫽0, and␣5⫽0. Since vector fields ⌳1共adS₀兲P

and⌳2共adS₀兲P generate an infinite dimensional Lie algebra, then case 共iii兲 also fails to exist.

In case 共iv兲, consider

⌳1共␭兲 = ⌳共␭兲 ␭ −␤1 and ⌳2共␭兲 = ⌳共␭兲 ␭ −␤2 .

We have⌳1共S0兲c共w兲=␣1e␤1w+␣2and⌳2共S0兲c共w兲=␣3e␤2w+␣4, where ␣1⫽0,␣2,␣3⫽0, and ␣4

are some constants. Note that

⌳1共adS₀兲P =␣1

冉

兺

j=−⬁ ⬁ e␤1wj ⳵ ⳵wj

冊

+␣₂S₀ and ⌳₂共adS₀兲P =␣3

冉

兺

j=−⬁ ⬁ e␤2wj ⳵ ⳵wj

冊

+␣₄S₀, and vector fields 兺_j=−⬁ _⬁e␤1wj共⳵/⳵_w

j兲 and 兺⬁j=−⬁e␤2 w_j_共_⳵_/_⳵

wj兲 generate an infinite dimensional Lie

algebra if␤1+␤2⫽0.

It follows from共i兲–共iv兲 that c共w兲 is one of the following forms: 共1兲 c共w兲=c2+ c3e␭w+ c4e−␭w,␭⫽0.

共2兲 c共w兲=c2+ c3w + c4w2, where c2− c4 are some constants.

䊐 Lemma 7: If the Lie algebra generated by the vector fields S0=兺⬁j=−⬁⳵/⳵wj,

Q =兺_j=−⬁ _⬁q共wj兲共⳵/⳵wj兲 , and S1=兺⬁j=−⬁兵˜␳j+ b˜共wj兲其共⳵/⳵wj兲 is of finite dimension, then q共w兲 is a

constant function.

Proof: It follows from Lemma 6 that 共1兲 q共w兲=c2+ c3w + c4w2 or

共2兲 q共w兲=c2+ c3e␭w+ c4e−␭w,␭⫽0,

where c2− c4 are some constants.

Consider case共1兲. We have

关S0,Q兴 = c3

兺

j=−⬁ ⬁ ⳵ ⳵wj + 2c4

兺

j=−⬁ ⬁ wj ⳵ ⳵wj = c3S0+ 2c4

兺

j=−⬁ ⬁ wj ⳵ ⳵wj . If c4⫽0, then 兺j=−⬁ ⬁wj共⳵/⳵wj兲苸Lxand兺j=−⬁ ⬁w2j共⳵/⳵wj兲苸Lx. If c₄= 0 and c₃⫽0, then 兺_j=−⬁ _⬁wj共⳵/⳵wj兲=共1/c3兲共Q−c2S0兲苸Lx.

If c₃= c₄= 0, then q共w兲=c₂ and there is nothing to prove.

Assume c₄2+ c₃2⫽0. Denote by P=兺⬁_j=−_⬁wj共⳵/⳵wj兲. Construct the vector fields

P₁=关P,S₁兴, Pn=关P,Pn−1兴, n ⱖ 2.

We have

DS0D−1= S0,

DS1D−1= S1−共ew− c˜兲S0,

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DP1D−1= P1+共− wew+ ew− c˜兲S0,

DP2D−1= P2+共− w2ew+ wew− ew+ c˜兲S0.

In general,

DPnD−1= Pn+共− wnew+ Rn−1共w兲ew+ cn兲S0, nⱖ 3,

where R_n−1 is a polynomial of degree n − 1 and cn is a constant. Since Lxis of finite dimension,

then there exists a natural number N such that

PN+1=␮NPN+ ¯ +␮1P1+␮0S0,

and S₀, P₁, . . . , PN are linearly independent. Thus

DPN+1D−1= D共␮N兲DPND−1+ ¯ + D共␮1兲DP1D−1+ D共␮0兲S0, or the same, ␮NPN+ ¯ +␮1P1+␮0S0+共− wN+1ew+ RN共w兲ew+ cN+1兲S0 = D共␮N兲兵PN+共− w N ew+ R_N−1共w兲ew+ cN兲S0其 + ¯ + D共␮1兲兵P1+共− we w + ew− c˜兲S₀其 + D共␮0兲S0.

By comparing the coefficients before PN, . . . , P1we have

␮N= D共␮N兲, ... ,␮1= D共␮1兲,

which implies that␮N, . . . ,␮1are all constants. By comparing the coefficients before S0we have

␮0− wN+1ew+ RN共w兲ew+ cN+1=␮N共− wNew+ RN−1共w兲ew+ cN兲 + ¯ +␮1共− wew+ ew− c˜兲

+ D共␮0兲.

The last equality shows that D共␮0兲−␮0is a function of w only. Thus D共␮0兲−␮0is a constant; we

denote it by d₀. The last equality becomes a contradictory one:

wN+1ew= RN共w兲ew+ cN+1−␮N共− wNew+ RN−1共w兲ew+ cN兲 − ¯ −␮1共− wew+ ew− c˜兲 − d0.

This contradiction proves that c₃2+ c₄2= 0, i.e., c3= c4= 0 in case 共1兲. Therefore, q共w兲=c2.

Consider case共2兲. Since

关S0,Q兴 = ␭c3

兺

j=−⬁ ⬁ e␭wj ⳵ ⳵wj −␭c4

兺

j=−⬁ ⬁ e−␭wj ⳵ ⳵wj , 关S0,关S0,Q兴兴 = ␭2c3

兺

j=−⬁ ⬁ e␭wj ⳵ ⳵wj +␭2c4

兺

j=−⬁ ⬁ e−␭wj ⳵ ⳵wj ,

then vector fields Q_␭= c3兺j=−⬁ ⬁e␭wj共⳵/⳵wj兲 and Q−␭= c4兺⬁j=−⬁e−␭wj共⳵/⳵wj兲 both belong to Lx. We

have DQ_␭D−1= Q_␭and DQ_−␭D−1= Q_−␭. Assume c3⫽0. Construct vector fields

Q₁=关Q_␭,S₁兴, Qn=关Q␭,Qn−1兴, n ⱖ 2.

Direct calculations show that

DQ₁D−1_{= Q}

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DQ2D−1= Q2− c3

2_{共1 + ␭兲e}共1+2␭兲w_S

0+ 2␭c3e共1+␭兲wQ␭.

It can be proved by induction on n that

DQnQ−1= Qn− pnS0+ qnQ␭, nⱖ 2, where pn= c3 n_{共1 + ␭兲共1 + 2␭兲 ¯ 共1 + 共n − 1兲␭兲e}_{共1+n␭兲w} , qn= nc3 n−1_{␭共1 + ␭兲 ¯ 共1 + 共n − 2兲␭兲e}共1+共n−1兲␭兲w_.

Since Lxis of finite dimension, there exists such a natural number N that

QN+1=␮NQN+ ¯ +␮1Q1+␮␭Q␭+␮0S0,

and S0, Q␭, Q1, . . . , QNare linearly independent. Then

DQN+1D−1= D共␮N兲DQND−1+ ¯ + D共␮0兲DS0D−1

or

␮NQN+ ¯ +␮1Q1+␮␭Q␭+␮0S0− pN+1S0+ qN+1Q␭= D共␮N兲兵QN− pNS0+ qNQ_␭其

+ ¯ + D共␮1兲兵Q1− p1S0+ q1Q␭其 + D共␮␭兲Q␭+ D共␮0兲S0.

By comparing the coefficients before QN, . . . , Q1, we have that ␮k, 1ⱕkⱕN, are all constants.

Comparing coefficients before S0 gives

␮0− pN+1= −␮NpN− ¯ −␮2p2−␮1p1+ D共␮0兲. 共41兲

Since pk, 1ⱕkⱕN+1, depend on w only, then D共␮0兲−␮0 is a function of w, and therefore

D共␮0兲−␮0is a constant; we denote it by d0.

If␭⫽−1/r for all r苸N, then pk⫽0 for all k苸N, and Eq.共41兲fails to be true.

Consider the case when ␭=−1/r for some r苸N. Substitution uj= e−␭wj transforms vector

fields共−1/␭c3兲Q␭,共−1/␭兲S1, and 共−1/␭兲S0 into vector fields

Q_␭ⴱ=

兺

j=−⬁ ⬁ ⳵ ⳵uj , S₁ⴱ=

兺

j=−⬁ ⬁ 兵˜␳ⴱ_j+ b˜ⴱ共uj兲其uj ⳵ ⳵uj , S₀ⴱ=

兺

j=−⬁ ⬁ uj ⳵ ⳵uj , where ␳ ˜ⴱ_j=

冦

⬘

共uj兲其

⳵ ⳵uj , Kª 1 2关Q␭ ⴱ_{,T兴 =}

兺

j=−⬁ ⬁ 兵j + c共uj兲其 ⳵ ⳵uj , where c共uj兲=b˜ⴱ

⬘

共uj兲+

j=−⬁ −1 aj ⳵ ⳵uj , nⱖ 1.

Since 兵Tn其n=1⬁ is an infinite sequence of linearly independent vector fields from Lx, then case r

= 1 fails to exist.

Consider case rⱖ2. We have

adQ ␭ ⴱS₁ⴱ=关Q_␭ⴱ,S₁ⴱ兴 =

兺

j=−⬁ ⬁

再

sgn共j兲r

冉

兺

k=0 j−1 uk r−1

冊

uj+˜␳jⴱ+ b˜ⴱ共uj兲 + uj˜bⴱ

⬘

共uj兲

冎

⳵ ⳵uj and ad_Q ␭ ⴱ r S₁ⴱ=

兺

j=−⬁ ⬁

再

r ! juj+ sgn共j兲r !

兺

k=0 j−1 uk+ d共uj兲

冎

for some function d,

ad_Q ␭ ⴱ r+1 S₁ⴱ=

兺

j=−⬁ ⬁ 兵2r ! j + d

⬘

共uj兲其 ⳵ ⳵uj . Note that vector fields ad_Q

␭ ⴱ r S₁ⴱand ad_Q ␭ ⴱ r+1

S₁ⴱhave coefficients of the same kind as vector fields T and K 共from case r=1兲 have. It means that ad_Q

␭ ⴱ r S₁ⴱ and ad_Q ␭ ⴱ r+1

S₁ⴱ generate an infinite dimensional Lie algebra. This contradiction implies that case rⱖ2 also fails to exist.

Thus, c3= 0. By interchanging ␭ with −␭, we obtain that c4= 0 also. Hence c3= c4= 0 and

q共w兲=c2. 䊐

We already know that a共␶兲=c0␶+ c1. The next lemma shows that c0⫽0.

Lemma 8: c₀is a nonzero constant.

⳵ ⳵␶j . Since 关T˜1,关T˜1,T˜ 兴兴 = 1 mc1j=−

兺

⬁ ⬁ b

⬙

共␶j兲 ⳵ ⳵␶j

and T˜1both belong to a finite dimensional Lx, then, by Lemma 6,共1兲 b

⬙

共␶兲=C˜1+ C˜2e␭␶+ C˜3e−␭␶or

共2兲 b

⬙

共␶兲=C˜1+ C˜2␶+ C˜3␶2for some constants C˜1− C˜3.

j=−⬁ ⬁ ⳵ ⳵␶j

produce an infinite dimensional Lie algebra Lxfor any fixed constants C2− C5. One can prove it by

showing that Lxcontains vector fields兺⬁j=−⬁jk共⳵/⳵␶j兲 for all k=1,2,.... Note that

关T˜1,T˜2兴 =

兺

j=−⬁ ⬁ 共− j + C2+ 2C3␶j+ 3C4␶j2+ 4C5␶3j兲 ⳵ ⳵␶j .

There are four cases: 共a兲 C5⫽0, 共b兲 C5= 0 , C4⫽0, 共c兲 C5= C4= 0, C3⫽0, and 共d兲 C5= C4= C3

= 0. In case 共a兲, 关T˜1,关T˜1,关T˜1,T˜2兴兴兴 − 6C4T˜1=

兺

j=−⬁ ⬁ 24C5␶j ⳵ ⳵␶j = 24C5P1苸 Lx, P1=

兺

j=−⬁ ⬁ ␶j ⳵ ⳵␶j ,

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⳵ ⳵␶j . Note that the Lie algebra generated by the vector fields

T ˜ 2 ⴱ_{= T}˜ 2−

冉

C3␶2− 1 2␶

冊

T ˜ 1= d共␶,␶1兲 ⳵ ⳵␶1 − d共␶−1,␶兲 ⳵ ⳵␶−1 +共d共␶,␶1兲 + d共␶1,␶2兲兲 ⳵ ⳵␶2 +¯ and

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T ˜ 1=

兺

j=−⬁ ⬁ ⳵ ⳵␶j

is infinite dimensional. It can be proved by comparing this algebra with the infinite dimensional characteristic Lie algebra of the chain

t1x= tx+ C3共t1 2

− t2兲 −12共t1+ t兲. 共42兲

Indeed, the Lie algebra Lx1 for 共42兲 is generated by operators 共18兲and 共19兲with d共t,t1兲=C3共t1 2

− t2_兲−1

2共t1+ t兲. To keep standard notations we set a共␶兲=−2C3␶− 1 and b共␶兲=C3␶2+ 1

2␶. Note that

since C₃⫽0, function a共␶兲 is not a constant. It follows from Theorem 3 proved below that the characteristic Lie algebras Lx 共and therefore algebra Lx1兲 for Eq. 共42兲 is of infinite dimension.

Thus, in case共c兲共ii兲 we also have an infinite dimensional Lie algebra Lx.

In case 共d兲, T ˜ 2=

兺

j=−⬁ ⬁ 共−␶−␶₁− ¯ −␶j−1+ C2␶j兲 ⳵ ⳵␶j 苸 Lx. Then J1= c2˜T1−关T˜1,T˜2兴 =

兺

j=−⬁ ⬁ j ⳵ ⳵␶j 苸 Lx and J₂= − 2

冉

关J₁,T˜₂兴 −

冉

1 2+ C2

冊

J1

冊

=_j=−

兺

_⬁ ⬁ j2 ⳵ ⳵␶j 苸 Lx.

Assuming that Jk, 1ⱕkⱕn, belong to Lx, by considering 关Jn, T˜2兴 one may show that Jn+1

=兺⬁_j=−_⬁jk+1_共_⳵_/_⳵␶

j兲苸Lx. It implies that Lxis of infinite dimension. 䊐

Let us introduce new variables,

wj= ln

冉

␶j+

c₁ c₀

冊

. Vector fields T₁and T in variables wjcan be rewritten as

兺

k=0 j−1 共ew_k − c˜兲 if jⱖ 1 0 if j = 0, −

兺

k=j −1 共ew_k − c˜兲 if j ⱕ − 1.

冧

c ˜ =c1 c0 , ˜共wb j兲 = − 1 m

冉

b共␶j兲 c0␶j+ c1

冊

We have DS0D−1= S0, DS1D−1= S1−共ew− c˜兲S0.

These lemmas allow one to prove the following theorem. Theorem 4: If the equation

t1x= tx+ a共␶兲tm+ b共␶兲tm−1+ ¯ , m ⱖ 1,

admits a nontrivial x -integral, then

共1兲 a共␶兲=c0␶and b共␶兲=c2␶2+ c3␶, where c0 , c2 , and c3are some constants.

共2兲 m=1 .

Proof: Consider case 共1兲. Define vector field

Q =关S₀,关S₀,S₁兴兴 − 关S₀,S₁兴 =

兺

j=−⬁ ⬁ 共b˜ 共w

⬙

j兲 − b˜ 共w

⬘

j兲兲 ⳵ ⳵wj .

By Lemma 7, b˜ 共w兲−b

⬙

˜ 共w兲=C for some constant C. Thus, b˜共w兲=C

⬘

0+ C1ew+ C2w for some

con-stants C₁, C₂, and C₀. Consider vector fields

jⱖ0 兵ew_j_共C 1C2 n wj n + P_n,j兲 + r_n,j共w,w₁, . . . ,w_j−1兲其 ⳵ ⳵wj +

兺

jⱕ−1 兵ew_j_共共C 1− 1兲C2 n wj n + Pn,j兲 + rn,j共w−1,w−2, . . . ,wj+1兲其 ⳵ ⳵wj ,

where Pn,j= Pn,j共wj, j兲 is a polynomial of degree n−1 whose coefficients depend on j; rn,jare the

functions that do not depend on wj. Since all vector fields Rn belong to a finite dimensional Lie

algebra Lx, then C1C2=共C1− 1兲C2= 0, or the same, C2= 0. Therefore,

b

˜共w兲 = C0+ C1ew.