Almost integrable evolution equations

Peter H. van der Kamp & Jan A. Sanders

Faculty of Sciences Division of Mathematics & Computer Science Vrije Universiteit, Amsterdam

Abstract

We present a 2-vector equation with exactly two symmetries, a counter example to Fokas’ conjecture that equations with as many symmetries as their dimension are integrable. Furthermore we prove the existence of infinitely many evolution equations with finitely many symmetries. We introduce the concept of almost integrability to describe the situation where one has a finite number of symmetries. The symbolic calculus of Gel’fand-Dikiĭ and p-adic analysis are used to prove our results.

1 Introduction

The use of symmetries in the study of differential equations was initiated by S. Lie. A Lie point symmetry transforms the dependent and independent variables such that the solution set remains invariant. This idea was generalized by E. Noether. She allowed the infinitesimal symmetry to depend on derivatives of the dependent variables, and proved a one to one correspondence between these generalized symmetries and conservation laws in the presence of a nondegenerate Lagrangian [12].

The nonlinear evolution equation u_{t} = u_{3} +u_{0}u_{1} (where we use the abbreviation u_{n} for the n^{th} derivative of u with respect to x) was derived to describe water waves in shallow canals by Korteweg and de Vries [10]. The equation was rediscovered by Kruskal and Zabusky and numerical computations verified the existence of local and stable waves. A bit of analytical understanding came with the discovery of conservation laws other than the ones governing energy and momentum; infinitely many were found in [11]. Inspired by the result of Noether infinitely many commuting symmetries of the KdV equation were constructed as well, see [7] and [13]. The existence of infinitely many conservation laws was viewed as integrability, a generalization of this concept for Hamiltonian systems in finite dimensional manifolds. Since then, several notions of integrability have been introduced, and one cannot even begin to describe the literature of this still expanding field. The relations among these different kinds of integrability are are strongly supported by computational evidence and sometimes by theorems. In this paper we say that a system is integrable if it has infinitely many generalized symmetries. A good introduction to this subject and the general reference is [14]. The background for some of the techniques we are using in this paper can be found in [16, 20].

In 1980 an observation was made at least twice by different authors. In [6] it is written

Another interesting fact regarding the symmetry structure of evolution equations is that in all known cases the existence of one generalized symmetry implies the existence of infinitely many.

and in [9] the same statement is made together with the footnote

This is not true for systems of equations. For example, the system u_{t} = u_{2}+(v^{2}/2), v_{t} = 2v_{2} has a nontrivial group, but this group is exhausted by the one-parameter (with parameter τ) group of transformations: u_{τ} = u_{3} + 3vv_{1}, v_{τ} = 4v_{3}.

However, although the remark in the footnote is true, the ’counter-example’ given there is an integrable system, cf. [1]. In spite of this fact Fokas adapted the remark and formulated the following conjecture in 1987.

Conjecture 1 (Fokas). If a scalar equation possesses at least one time-indepen-dent non-Lie point symmetry, then it possesses infinitely many. Similarly for nvector equations one needs n symmetries.

Four years later Bakirov [1] published the first example of a non-integrable equation in the possession of a generalized symmetry. This was a fourth order 2-vector equation with a sixth order symmetry and it was shown (with extensive computer algebra computations) that there are no other symmetries up to order 53. The authors of [3] proved in 1998 that the system of Bakirov does not posses another symmetry at any higher order, thereby proving that indeed one symmetry does not imply integrability.

For n = 1 the conjecture of Fokas is proven to be true for a large class of equations in [16]. In section 2 an outline of this proof is given, together with references to the development of the symbolic calculus.

In [18] it is stated that the Bakirov equation seems very exceptional. This is not the case, as we show in section 3 that there are infinitely many families of non-integrable 2-vector equations in the possession of nontrivial generalized symmetries. Therefore these kind of equations are as common (or as rare) as integrable equations. We propose to call them almost integrable. This terminology somehow reflects the idea of the conjecture.

Definition 1. An equation is called (symmetry-)integrable if it possesses infinitely many generalized symmetries and almost integrable of depth (at least, at most) n if there are exactly (at least, at most) n generalized symmetries. When an equation is almost integrable but not integrable we say that it is almost integrable of finite depth.

In section 4 we show the Fokas’ conjecture to be false for n = 2 by constructing a 6-parameter family of 2-vector equations which is almost integrable of depth 2.

These results are of importance both to the the somewhat philosophical question ”What is integrability?” and to the more practical question of how to algorithmically check integrability, since some authors take the conjecture as their definition of integrability (cf. [15]). It is our point of view that this last choice does not suffice, and that it is necessary to produce an explicit proof for the existence of infinitely many symmetries.

We felt it appropriate to give detailed proofs (appendix A). To convince the reader that the approach is largely constructive we have included our counter example and both symmetries explicitly in appendix C. Appendix B contains the Maple (cf. [5]) code that produces them.

2 A classification of scalar equations

The class of homogeneous scalar equations with non-negative scaling parameter of the form

u_{t} = u_{m} + f(u, u_{1}, …, u_{m−1}),

is classified with respect to symmetries in [16, 17]. To find equations with symmetries one usually derives obstructions by requiring the Lie-bracket to vanish. We indicate how Sanders and Wang were able to:

1. Deduce an obstruction for the general equation to posses a symmetry on an arbitrary order.

2. Reduce the number of obstructions to be solved.

The first is done in the symbolic calculus as developed by Gel’fand and Dikiĭ in [8]. See also [19] where the calculus was used in connection with symmetries and [20] where all the technical details are worked out. There is a one to one correspondence between functions like f and symmetric polynomials in the symbol ξ, i.e.

û_{i}u_{j} = \frac{ξ_{1}^{i}ξ_{2}^{j} + ξ_{1}^{j}ξ_{2}^{i}}{2} u^{2}.

In the symbolic calculus the obstructions become divisibility conditions, which can be solved for infinitely many orders at once.

For the second an implicit function theorem was formulated and proven. This theorem states that under certain technical conditions the existence of one symmetry ensures us that the number of obstructions to be solved for proving the existence of the other symmetries is finite and independent of the order of the symmetries. By a method based on techniques from Diophantine approximation theory it is proven in [2] that these technical conditions are satisfied. That all almost integrable equations appear to be integrable proves part of Fokas’ conjecture for n = 1.

3 B-equations

The class B consist of all B_{a}^{n}(K) which have the explicit form

( \begin{matrix}u_{t} \\ v_{t} \end{matrix} ) = ( \begin{matrix}au_{n} + K(v, v_{1}, …, v_{n−1}) \\ v_{n} \end{matrix} ) , a ∈ ℂ,

where K is a quadratic polynomial with complex coefficients and the order of the equation (n) exceeds the number of derivatives for each term of K. B contains a lot of integrable equations as well as almost integrable equations of finite depth. The property which makes B nice is that all symmetries of almost all B_{a}^{n}(K) are in B, cf. [1]. The only exceptions are a = 1 or n = 1, cf. [4].

In the symbolic calculus the first condition for B_{b}^{m}(S) to be symmetry of the B_{a}^{n}(K) is:

[B_{a}^{n}(K), B_{b}^{m}(S)] = 0 ⇔ G_{n}[a] · Ŝ = G_{m}[b] · K̂,  (1)

where the G-functions are defined as follows:

G_{n}[a](ξ) =^{def} a(ξ_{1} + ξ_{2})^{n} − ξ_{1}^{n} − ξ_{2}^{n}.

This is the only condition to be satisfied because we have that

[( \begin{matrix}K \\ 0 \end{matrix} ) , ( \begin{matrix}S \\ 0 \end{matrix} )] = 0

is satisfied when both K and S are quadratic in v and its derivatives.^{1} Consequently if we find some b and m such that G_{a}[n] divides the product G_{m}[b]K̂, then Ŝ is polynomial in ξ and we have found a symmetry explicitly.

Consider B_{a}^{4}(K). This system has a symmetry if G_{4}[a] divides G_{m}[b] · K̂. It is possible that some factor F̂ of G_{4}[a] divides K̂. Because K̂(ξ) is a sym-

metric polynomial of degree smaller than 4 (since the number of derivatives in each term of K is smaller than the order of the equation) and the only symmetric first degree polynomial ξ_{1} + ξ_{2} does not divide G_{4}[a](ξ) for any a (which rules out that deg(F̂(ξ)) equals 1 or 3) this factor F̂(ξ) should have degree 2. But then G_{4}[a]/F̂ can be written as cG_{2}[d] for some c, d. We see that in this case B_{a}^{4}(K) is in the hierarchy of B_{d}^{2}(K̂/cF̂),

1This makes B nice, we do not need the implicit function theorem here

G_{2}[d] · K̂ = G_{4}[a] · \frac{K̂}{cF̂} .

In the following, since we want to produce equations which are almost integrable of finite depth, we are going to search for all a for which there exist some b such that G_{4}[a] divides G_{m}[b] at some order m. Observe that for such pairs (a, 4) and (b, m), the result is valid for any (not necessarily homogeneous) K and thus the number of parameters in the family of almost integrable B_{a}^{4} equals six (the number of symmetric polynomials in two variables of degree less than four).

Because the G-functions are symmetric in ξ, we use

X =^{def} ξ_{1} + ξ_{2} and Y =^{def} ξ_{1}ξ_{2}

and remark that all symmetric functions can be written in terms of X and Y. It follows from lemma 3 that we can always find some polynomial B_{m}(c) such that Y − cX^{2} divides G_{m}[B_{m}(c)](X, Y). Therefore we write G_{4}[a] as a function of X, Y and factorize:

\begin{aligned}G_{4}[a](X, Y) & = & (a − 1)X^{4} + 4X^{2}Y − 2Y^{2} \\ & = & −2(Y − (1 + α)X^{2})(Y − (1 − α)X^{2}) \end{aligned}

with

α =^{def} \sqrt{\frac{a + 1}{2}}.

We see that G_{4}[a] divides G_{m}[B_{m}(1 ± α)] only if B_{m}(1 + α) equals B_{m}(1 − α). This condition is trivially satisfied when α = 0. We call the point where this happens (a = −1) the trivial root. It will be removed by the following definition.

Definition 2.

Q_{m}(α) =^{def} \frac{B_{m}(1 + α) − B_{m}(1 − α)}{α}.  (2)

By the results of appendix A the roots of Q_{m}(α) give the eigenvalues a of all almost integrable B_{a}^{4}(K).

Lemma 1. Q_{m}(α) has a nontrivial root if 5 < m ≠ 1 mod 3.

Proof. Lemma 4 proves that Q_{m}(α) is a polynomial in a of degree ⌊m/4 − 1/2⌋. 2 This means that Q_{m}(α) has a root for all m > 5. Lemma 5 proves that −1 is a root only if m = 1 mod 3. This is not surprising, the family B_{−1}^{4} is integrable and has symmetries at all orders 1 mod 3, cf. [4].

2The Entier function of x ∈ ℝ is notated as ⌊x⌋. It returns the largest z ∈ ℤ such that z ≤ x.

Theorem 1. There are infinitely many systems which are almost integrable of finite depth.

Proof. By lemma 1 we have infinitely many candidates, given by B_{a}^{4}(K) with a the non-trivial roots of Q_{m}(α). In principle all but a finite number of these roots could equal an eigenvalue of an integrable equation. This is not the case. By corollary 2 and the results of [4] we know all integrable B_{a}^{4}. Their eigenvalues are a = 0, −1, −3 and their symmetries appear at orders m = 4 mod 8, 1 mod 3, 0 mod 4 respectively. Now take for example roots of Q_{m}(a) for m = 6 mod 12. The corresponding equations have at least one symmetry and they are not integrable.

To end the section we remark that the Bakirov equation mentioned in the introduction is B_{5}^{4}(v^{2}) (its symmetry is B_{11}^{6}(5v_{2}v + 4v_{1}^{1})) and that finding all almost integrable B_{a}^{5}(K) can be done with the same method.

4 The counter example to Fokas’ conjecture

Theorem 2. There exists a 2-vector equation which is almost integrable of depth two.

Proof. There exist a, c ∈ ℂ such that

( \begin{matrix}u_{t} \\ v_{t} \end{matrix} ) = ( \begin{matrix}au_{7} + 7v_{0}v_{3} + cv_{1}v_{2} \\ −v_{7} \end{matrix} )

has exactly two symmetries. We first prove the existence of two symmetries at order 11 and 29. Second, we calculate the coefficients and show how to calculate the symmetries of the equation. Third, we prove that there is no other symmetry.

By lemma 6 every G-function of the form G_{7}[A_{7}(r)] with

A_{n}(r) =^{def} \frac{1 + r^{n}}{(1 + r)^{n}}

is divisible by some third degree polynomial. The remainder is a fourth degree symmetric polynomial for which we can determine whether it divides the higher G-functions by using the method described in section 3,

\frac{G_{7}[A_{7}(r)](ξ_{1}, ξ_{2})}{(ξ_{1} − rξ_{2})(rξ_{1} − ξ_{2})(ξ_{1} + ξ_{2})} = −7(1 + r)^{4}(Y − pX^{2})(Y − qX^{2})

with p(r), q(r) the solutions of

(1 + r)^{4}x^{2} − (1 + r)^{2}(2r^{2} + 3r + 2)x + (r^{2} + r + 1)^{2} = 0.

Like in section 2 there exists a symmetry on order m if B_{m}(p) equals B_{m}(q). The fact that (B_{11}(p) − B_{11}(q)) and (B_{29}(p) − B_{29}(q)) share the common (non-trivial) factor

P(r) =^{def} r^{6} + 7r^{5} + 19r^{4} + 25r^{3} + 19r^{2} + 7r + 1

proves the existence claim.

Take P(β) = 0. To obtain the eigenvalue of the equation we compute A_{7}(β),

a = 42β^{5} + 280β^{4} + 700β^{3} + 798β^{2} + 504β + 104.

To obtain the other coefficient c we translate

K̂ = \frac{7}{2}(ξ_{1} − βξ_{2})(ξ_{1} − \frac{1}{β} ξ_{2})(ξ_{1} + ξ_{2})v^{2}

back to differential language,

c = 7β^{5} + 49β^{4} + 133β^{3} + 175β^{2} + 126β + 56.

To obtain the eigenvalues of the symmetries take γ to be a root of G_{7}[a] not equal to −1, β, 1/β and compute A_{11}(γ) and A_{29}(γ). Finally one obtains the quadratic parts

Ŝ = \frac{G_{i}[A_{i}(γ)] · K̂}{G_{7}[a]} , i = 11, 29.

Now we prove that the equation is almost integrable of depth at most two. Let δ be a root of G_{7}[a] not equal to −1, β, 1/β, γ, 1/γ. If the equation has a symmetry on order k then we have

A_{k}(δ) = A_{k}(γ)  (3)

which can be written in the form

U_{k} =^{def} a^{k} + b^{k} − c^{k} − d^{k} = 0.

with

a = 1 + γ, b = (1 + γ)δ, c = 1 + δ, d = (1 + δ)γ.

Now the problem is to find all k for which U_{k} = 0. The following lemma was proven in [3] and used to prove that the Bakirov equation is almost integrable of depth one.

Lemma 2. (Skolem) Suppose p is an odd prime. Let a, b, c, d ∈ ℤ_{p} and suppose they are not zero modulo p. By the little theorem of Fermat we can find ã, b̃, c̃, d̃∈ ℤ_{p} such that

a^{p−1} = 1 + pã, b^{p−1} = 1 + pb̃, c^{p−1} = 1 + pc̃, d^{p−1} = 1 + pd̃,

Define

V_{k} =^{def} ãa^{k} + b̃b^{k} − c̃c^{k} − d̃d^{k}.

The following holds

• If U_{k} ≢ 0 mod p then U_{k+r(p−1)} ≠ 0 for all r ∈ ℤ.

• If U_{k} = 0 and V_{k} ≢ 0 mod p then U_{k+r(p−1)} = 0 implies r = 0.

From now on everything can be done p-adically. Due to the lemma of Skolem we only have to check a finite number (p − 1) of orders if all the involved roots are in ℤ_{p}. Moreover all the calculations can be done modulo p^{2}. For our purpose the prime

p =^{def} 101

suffices. We choose β to be a particular root of P(r) and calculate a:

β ≡ 30 + 86p → a ≡ 62 + 37p.

From the roots of G_{7}[a] we choose:

γ ≡ 68 + 100p, δ ≡ 96 + 13p.

This gives

\begin{matrix}a ≡ 69 + 100p & , b ≡ 59 + 58p & , c ≡ 97 + 13p & , d ≡ 31 + 44p \\ ã = 56 & , b̃ = 38 & , c̃ = 36 & , d̃= 94. \end{matrix}

For 0 ≤ k < p−1 we have U_{k} ≡ 0 mod p only if k = 0, 1, 7, 11, 29. Moreover for k = 0, 1, 7, 11, 29 we have V_{k} ≢ 0 mod p. Lemma 2 then states that k = 0, 1, 7, 11, 29 are the only values for which condition (3) can be satisfied.

Remark 1. Every root of P(r) gives rise to an almost integrable equation of depth at least two. Because the roots appear in pairs like β, \frac{1}{β} and A_{n}(r) is invariant under r → \frac{1}{r}, only three of these equations are different. For only one of them (by choosing β) we have proven it is an almost integrable equation of depth at most two as well.

Corollary 1. One still has some freedom in the choice of K̂. If one multiplies K̂ = v^{2} with any symmetric polynomial in ξ of degree less than 4 not divisible by Y −pX^{2} or Y −qX^{2}, the proof does not change, although the symmetries evidently do. This shows that instead of one example of a equation with 2 symmetries we have found a 6-parameter family.

A Some technical results

Lemma 3. For all c ∈ ℂ and 1 < m ∈ ℕ there exist b ∈ ℂ such that Y − cX^{2} divides G_{m}[b](X, Y). Proof. We write

R_{m}(ξ) =^{def} ξ_{1}^{m} + ξ_{2}^{m}

in terms of the symmetric polynomials X and Y.

R_{0} = 2, R_{1} = X, R_{m+2} = XR_{m+1} − Y R_{m}.  (4)

By substitution of Y = cX^{2} in (4) and induction on m we have

R_{m}(X, Y) = B_{m}(c) · X^{m}

for some polynomial B_{m}(c) of degree ⌊\frac{m}{2}⌋. One sees that G_{m}[b](X, Y) = bX^{m} − R_{m}(X, Y) is in the ideal of Y − cX^{2} if one takes b = B_{m}(c). Lemma 4. Q_{m}(α(a)) is a polynomial in a of degree ⌊\frac{m−2}{4}⌋. Proof. Consider

F = \frac{(1 + α)^{n} − (1 − α)^{n}}{α} .

Using the binomial formula we write F in terms of a as:

F = ∑_{k=0}^{⌊ \frac{n−1}{2} ⌋} 2^{k+1} ( \begin{matrix}n \\ 2k + 1 \end{matrix} ) (a + 1)^{k}.

By lemma 3 the highest power of the polynomial B_{m}(x) is n = ⌊\frac{m}{2}⌋. It follows that Q_{m}(α(a)) is polynomial in a and has degree ⌊\frac{⌊ \frac{m}{2} ⌋−1}{2}⌋. To see what is going on at the trivial root a = −1 of Q_{m} we need explicit formulas. Solving the recursion relation (4) gives

B_{n}(x) = (\frac{1 + \sqrt{1 − 4x}}{2})^{n} + (\frac{1 − \sqrt{1 − 4x}}{2})^{n}.  (5)

Lemma 5. a = −1 is a root of Q_{m}(α(a)) only if m = 1 mod 3. Proof. We look at the multiplicity of α in

P_{m}(α) =^{def} αQ_{m}(α).

Of course the multiplicity of α is at least 1. The explicit formula for the derivative of P_{m}(α) with respect to α in the point where a = −1 is

Pm^{′}(0) = \frac{m\sqrt{3}i}{6}((\frac{1 − i\sqrt{3}}{2})^{m−1} − (\frac{1 + i\sqrt{3}}{2})^{m−1}).

Recognize the primitive third roots of −1. One sees that Pm^{′}(0) = 0 only if m = 1 mod 3.

Lemma 6. • (ξ_{1} + ξ_{2}) divides G_{n}[a](ξ) for all odd n ∈ ℕ

• (ξ_{1} − rξ_{2})(rξ_{1} − ξ_{2}) divides G_{n}[A_{n}(r)](ξ) for all 1 < n ∈ ℕ and −1 ≠ r ∈ ℂ.

Proof. By substitution of ξ_{1} = −ξ_{2} in G_{n}[a](ξ) and substitution of ξ_{1} = rξ_{2}, ξ_{2} = rξ_{1} in G_{n}[A_{n}(r)](ξ).

We are going to look at the subclass of B where the eigenvalue is zero and will work with g-functions

g_{n}(\frac{ξ_{1}}{ξ_{2}} ) = 1 + (\frac{ξ_{1}}{ξ_{2}} )^{n} = −ξ_{2}^{n}G_{n}[0](ξ_{1}, ξ_{2})

in the new symbol \frac{ξ_{1}}{ξ_{2}} .

Lemma 7. For all n the family B_{0}^{n} is integrable. There exists symmetries at order n mod 2n.

Proof. There exists a symmetry B_{0}^{m}(S) if g_{n} divides g_{m}. Its quadratic part is given by

Ŝ = ξ_{2}^{m−n} \frac{g_{m}(\frac{ξ_{1}}{ξ_{2}} )}{g_{n}(\frac{ξ_{1}}{ξ_{2}} )} · K̂.

Let x = \frac{ξ_{1}}{ξ_{2}} . We have that x^{(2k+1)n} = −1 whenever x^{n} = −1.

Corollary 2. In particular the family B_{0}^{4} is integrable with symmetries at order 4 mod 8.

B MAPLE code to compute the counter example and its symmetries

G:=unapply(a*(x+1)^n-x^n-1,a,n):
A:=unapply(factor((1+r^n)/(1+r)^n),r,n):
alias(alpha=RootOf(x^6+7*x^5+19*x^4+25*x^3+19*x^2+7*x+1)):
a.7:=factor(A(alpha,7)):
K:=expand(7*(x-factor(1/alpha))*(x+1)*(x-alpha)):

for i from 0 to 3 do
   c.7.c.i:=factor(coeff(K,x,i))
od:
P:=factor(G(A(alpha,7),7)/K):
beta:=3+18*alpha+32*alpha^2+31*alpha^3+13*alpha^4+2*alpha^5:
for p in [11,29] do
   a.p:=factor(A(beta,p)):
   F:=factor(G(a.p,p)):
   for i from 0 to (p-5)/2 do
      c.p.c.i:=factor(coeff(expand(factor(F/P)),x,i)):
   od
od:
for p in [7,11,29] do
   K.p:=a.p*u[p]:
   for i from 0 to (p-5)/2 do
      K.p:=K.p+c.p.c.i*v[i]*v[p-4-i]
   od:
   print([K.p,v[p]])
od:

C Explicit expressions

The counter example, with α^{6} + 7α^{5} + 19α^{4} + 25α^{3} + 19α^{2} + 7α + 1 = 0.

u_{t} = −(42α^{5} + 280α^{4} + 700α^{3} + 798α^{2} + 504α + 104)u_{7} +
+(7α^{5} + 49α^{4} + 133α^{3} + 175α^{2} + 126α + 56)v_{1}v_{2} +
+7v_{3}v_{0}
v_{t} = v_{7}.

The symmetry at order 11:

u_{t} = (220α^{5} + 1474α^{4} + 3718α^{3} + 4312α^{2} + 2772α + 617)u_{11} +

(1452α^{5} + 9394α^{4} + 22198α^{3} + 22440α^{2} + 12276α + 803)v_{4}v_{3} + (308α^{5} + 2002α^{4} + 4774α^{3} + 4928α^{2} + 2772α + 275)v_{6}v_{1} + (891α^{5} + 5775α^{4} + 13695α^{3} + 13959α^{2} + 7722α + 605)v_{5}v_{2} +

(44α^{5} + 286α^{4} + 682α^{3} + 704α^{2} + 396α + 44)v_{7}v_{0}
v_{t} = v_{11}.

Almost integrable evolution equations 12 The symmetry at order 29:

u_{t} = (448920α^{5} + 3003240α^{4} + 7555080α^{3}

+8717400α^{2} + 5574960α + 1214521)u_{29}

+ (270128069783α^{5} + 1727757945945α^{4} + 3990463528125α^{3} +

+3816707978927α^{2} + 1925811490446α − 49827643081)v_{14}v_{11}

+ (314433121731α^{5} + 2010669255777α^{4} + 4641691138509α^{3} +

+4434301309155α^{2} + 2233269457038α − 62591719089)v_{12}v_{13}

+ (198976380529α^{5} + 1273247762337α^{4} + 2943442920489α^{3} +

+2821845288637α^{2} + 1429010624934α − 30989458696)v_{10}v_{15}

+ (29965743529α^{5} + 192236235213α^{4} + 446681340621α^{3} +

+433712137405α^{2} + 223951932702α + 110178395)v_{18}v_{7}

+ (11152408954α^{5} + 71626741966α^{4} + 166814925142α^{3} +

+162888051034α^{2} + 84821188356α + 845554160)v_{19}v_{6}

+ (66797105050α^{5} + 428083102004α^{4} + 992668562528α^{3} +

+958988226022α^{2} + 491408490672α − 4019417545)v_{8}v_{17}

+ (125159366438α^{5} + 801426426464α^{4} + 1855203992108α^{3} +

+1784579666114α^{2} + 908464101048α − 14251495839)v_{9}v_{16}

+ (813186274α^{5} + 5236057532α^{4} + 12256814504α^{3} +

+12117221902α^{2} + 6424917984α + 192844664)v_{21}v_{4}

+ (3380780125α^{5} + 21740156211α^{4} + 50757689727α^{3} +

+49864019173α^{2} + 26198558298α + 521618795)v_{20}v_{5}

+ (149249573α^{5} + 962305289α^{4} + 2258649833α^{3} +

+2247288329α^{2} + 1202541318α + 48142146)v_{22}v_{3}

+ (19626127α^{5} + 126714195α^{4} + 298215555α^{3} +

+298616683α^{2} + 161233794α + 8024996)v_{23}v_{2}

+ (66120α^{5} + 428040α^{4} + 1012680α^{3} +

+1026600α^{2} + 563760α + 38280)v_{25}v_{0}

+ (1645982α^{5} + 10641492α^{4} + 25110984α^{3} +

+25302674α^{2} + 13780800α + 814349)v_{24}v_{1}

v_{t} = v_{29}.

References

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[2] Frits Beukers. On a sequence of polynomials. Journal of Pure and Applied Algebra, 117 & 118:97–103, 1997.

[3] Frits Beukers, Jan A. Sanders, and Jing Ping Wang. One symmetry does not imply integrability. J. Differential Equations, 146(1):251–260, 1998.

[4] Frits Beukers, Jan A. Sanders, and Jing Ping Wang. On integrability of systems of evolution equations. Journal of Differential Equations, 172(2):396– 408, 2001.

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