Nonintegrabilty of a Halphen system

We study the Halphen system with real variables and real constants. We show that in the case where at least one constant is nonzero, this system does not admit any first integral that can be described by formal power series. It hence follows that analytic first integrals do not exist. Furthermore, we prove that first integrals of the Darboux type also do not exist.


Introduction to the problem
We consider the systeṁ where x 1 , x 2 , and x 3 are real variables and α 1 , α 2 , and α 3 are real constants. We call system (1) the second Halphen system (Halphen himself called it the second system [1]) because system (1) with α 1 = α 2 = α 3 = 0 becomes the so-called classical Halphen system. The classical Halphen system is a famous model (see, e.g., [1]- [3]), which first appeared in Darboux's work [1] and was later solved by Halphen [3]. One of the circumstances making this system famous is that this system, as was shown, is equivalent to the Einstein field equations for a diagonal self-dual Bianchi-IX metric with a Euclidean signature (see [2], [4]). The classical Halphen system also arises in similarity reductions of associativity equations on a three-dimensional Frobenius manifold [5]. From the standpoint of integrability, the classical Halphen system has been intensively studied using different theories. One of the main results in this direction is that system (1) with α 1 = α 2 = α 3 = 0 can be explicitly integrated because we can express its general solution in terms of elliptic integrals (see [3], [6], [7]), but the first integrals are not global and are multivalued nonalgebraic functions (see [8]). Other results that we mention are in [9], where the so-called Darboux polynomials were used to prove that system (1) with α 1 = α 2 = α 3 = 0 does not admit a nonconstant algebraic first integral, and finally in [10], where a complete characterization of the formal and analytic first integrals was provided.
One of the ideas that we use to characterize the existence of formal first integrals is as follows. The three planes are invariant under the flows of system (1), and if f : is a first integral of system (1), then the restriction of f to H i = 0 for each i = 1, 2, 3 is also a first integral of system (1). Hence, the method for proving our results consists of completely studying the integrability of the reduced systems on each H i = 0 to obtain exact information on the integrals of the whole system (1).
The main results in this paper are as follows. (1) is a nonconstant formal power series in the variables x 1 , x 2 , and x 3 such that (1) does not admit any formal first integral.
Here an analytic first integral of system (1) is a nonconstant analytic function that is constant over the trajectories of system (1). We obtain the following result as a corollary of Theorem 1. (1) does not admit any analytic first integral.

Theorem 2. For any
A rational first integral f = f (x 1 , x 2 , x 3 ) of system (1) is a nonconstant rational function that is constant over solutions of system (1).
Finally, we study the Darboux first integrals of system (1) (see below for the definition). This paper is organized as follows. In Sec. 2, we present some auxiliary results needed for proving Theorems 1 and 2. In Sec. 3, we prove Theorems 1 and 2. In Sec. 4, we prove Theorem 3. Finally, in Sec. 5, we prove Theorem 4.

Auxiliary results
We note that system (1) is invariant under the changes The first auxiliary result can be easily proved using Newton's binomial formula, and we omit its proof (see [10] for a proof).

Lemma 1.
Let f = f (x 1 , x 2 , x 3 ) be a formal power series such that at x l = x j , l, j ∈ {1, 2, 3}, l = j, we have f (x 1 , x 2 , x 3 )| x l =xj =f , wheref is a formal power series in the variables x j , x k , k ∈ {1, 2, 3}, k = j and k = l. Then there exists a formal series g = g( We recall the definition of a Darboux polynomial for system (1) with a cofactor K. We say that Furthermore, because the polynomials in the right-hand side of (1) have degree two, the cofactor K has a degree at most one. We write it as We note that f = 0 is an invariant algebraic surface for the flow of system (1), and a polynomial first integral of system (1) is a Darboux polynomial with a zero cofactor. We recall that if there exist invariant planes under the flows of system (1) and a Darboux polynomial of system (1) with a cofactor K, then the restriction of f to each of the invariant planes is a Darboux polynomial of system (1) restricted to each of the planes and with cofactors being the restriction of K to each of the planes. We note that for real polynomial differential systems such as system (1), when we seek their Darboux first integrals, we use complex Darboux polynomials and complex exponential factors in the general case because these objects appear in pairs (of them and their conjugates), which forces the Darboux first integral to become real whenever it exists.
The following two lemmas are well-known.
if we write f as a sum of its homogeneous parts, f = f 1 + · · · + f n , then f is a Darboux polynomial of system (1) with a cofactor K if and only if f i is a Darboux polynomial of system (1) with a cofactor K for all i = 1, . . . , n.

Lemma 4. Let F be an analytic function and F = i F i be its decomposition into homogeneous polynomials of degree i. Then F is an analytic first integral of homogeneous polynomial differential system (1) if and only if F i is a homogeneous polynomial first integral of system (1) for all i.
The proofs of Lemmas 2-4 use the homogeneity of system (1). Their proofs are well known (see, e.g., [13] for the first two and [14] for the third).
As usual, N denotes the set of positive integers. Lemma 5 was proved, for instance, in [13].

Lemma 5. If we decompose the polynomial f into its irreducible factors in
The following statement is important for investigating the rational integrability of polynomial systems. It was proved in [15]. Proposition 1. The existence of a rational first integral for polynomial differential system (1) implies either the existence of a polynomial first integral (and hence a Darboux polynomial with a zero cofactor) or the existence of two coprime Darboux polynomials with the same nonzero cofactor.
An exponential factor F of polynomial differential system (1) for some polynomial L of degree one. The following result is well known. Its proof and geometric meaning were given in [16] and [17].

Proposition 2.
The following statements hold: is an exponential factor for polynomial system (1) and g 1 is not a constant polynomial, then g 1 = 0 is an invariant algebraic curve.

2.
Eventually, e g0 can be exponential factors, coming from the multiplicity of the infinite invariant straight line.
The following result given in [16] characterizes the algebraic multiplicity of an invariant algebraic surface using the number of exponential factors of system (1) associated with this invariant algebraic surface.

Theorem 5.
Given an irreducible invariant algebraic surface g 1 = 0 of degree m of system (1), it has the algebraic multiplicity k if and only if the vector field associated with system (1) has k−1 exponential factors of the form e g0,i/g i 1 , where g 0,i is a polynomial of degree at most im and g 0,i and g 1 are coprime for i = 1, . . . , k − 1.
In view of Theorem 5, if we prove that e g0/g1 is not an exponential factor with a degree g 0 not exceeding the degree g 1 , then there are no exponential factors associated with the invariant algebraic surface g 1 = 0.
A first integral G of system (1) is of the Darboux type if it has the form where f 1 , . . . , f p are Darboux polynomials, F 1 , . . . , F q are exponential factors, and λ j , μ k ∈ C for j = 1, . . . , p and k = 1, . . . , q.
We need the following result, whose proof was given in [13].
Theorem 6. Let system (1) admit p Darboux polynomials with cofactors K i and q exponential factors if and only if the function G given in (6) (of the Darboux type) is a first integral of system (1).
As already noted in the introduction, the planes x 1 = x 2 , x 1 = x 3 , and x 2 = x 3 are invariant under flows of (1). Therefore, if f is a formal first integral of system (1), then are formal first integrals of system (1) restricted to the respective planes x 1 = x 2 , x 1 = x 3 , and x 2 = x 3 .
where c 0 is some constant and g := g(x 1 , x 2 , x 3 ) is a formal power series.
Proof. Let f be a formal first integral of system (1). We first prove that f 1 = c 0 , where f 1 is given in (7). Indeed, f 1 satisfies We now introduce the linear change of variables In these new variables, we have We show that h = c 0 . For this, we write h as a power series in y 2 and z 2 , Hence, requiring that h satisfies (9), we obtain where h m,n = 0 for m < 0 or n < 0. Computing the different degrees in y 2 and z 2 in (10), we now obtain We claim that h k,l = 0 for k, l ≥ 0 if (k, l) = (0, 0).
For f 2 , repeating the arguments that we applied for f 1 , we now find that there exists a constant c 1 and a formal power series g 1 Further, repeating the same arguments for f 3 , we find that there exists a constant c 2 and a formal power Substituting x 1 = x 2 = x 3 = 0 in Eqs. (13)-(15), we now obtain c 0 = c 1 = c 2 . From (13)-(15), we also obtain ( which clearly implies that there exists a formal power series g := g(x 1 , x 2 , x 3 ) such that The proposition now follows from (13) and the first relation in (16).

Proof of Theorems 1 and 2
Proof of Theorem 1. Let f be any formal first integral of system (1). By Proposition 3, we know that f can be written as for some constant c 0 and some formal power series g := g(x 1 , x 2 , x 3 ). Requiring that f be a first integral of system (1) and simplifying by ( where the derivative is evaluated along a solution of system (1). We prove that g = 0. For this, we proceed by reduction to the absurd: we suppose g = 0 and reach a contradiction. We consider two different cases.
We now suppose that (20) holds for j = 0, . . . , m − 1 with m ≥ 1 and prove it for k = m. Clearly, by the induction hypothesis, we have and then from (19), after dividing by z m 2 , we obtain Evaluating this equation at z 2 = 0, we then obtain Because h 0,m is a formal series in y 2 , we have c m = 0 and hence h 0,m = 0, which proves (20) for j = m. By the induction hypothesis, (20) then holds for all j ≥ 0, and we obtain h 0 = 0 from (20). Hence g 0 = 0, in contradiction with the fact that g is not divisible by

Case 2.
In the case where g is divisible by x 1 − x 2 , g = (x 1 − x 2 ) j h with j ≥ 1 and h = 0 and h := h(x 1 , x 2 , x 3 ) is a formal power series that is not divisible by x 1 − x 2 and after dividing by (x 1 − x 2 ) j , we find that the series satisfies the equation where the derivative of h is evaluated along a solution of system (1). Applying the arguments to h similar to those used for g in Case 1, we conclude that h = 0, thus obtaining a contradiction.
Hence, g = 0, and the proof of the theorem follows from (17) and the definition of a formal first integral.
Proof of Theorem 2. By Theorem 1, Halphen system (1) has no polynomial first integrals. The proof of Theorem 2 now follows immediately from Lemma 4.

Proof of Theorem 3
We recall that the equation defining a Darboux polynomial is given in (4) and that in view of Lemma 2, we can take We prove Theorem 3 using Proposition 1, Theorem 1, and the following result. Theorem 7. For (α 1 , α 2 , α 3 ) ∈ R 3 \ {(0, 0, 0)}, every Darboux polynomial of system (1) has the form where c is some constant and n 1 , n 2 , and n 3 are nonnegative integers. Furthermore, the cofactor of f is Our main objective in this section is to prove Theorem 7 because, as becomes clear later, it easily implies the proof of Theorem 3. For this, we study the Darboux polynomials of system (1) of degree one and of a degree greater than one. We do this in two separate propositions.
To prove this proposition, we show that each of the coefficients c 1 , c 2 , and c 3 in the definition of K given in Lemma 2 is zero for any Darboux polynomial of system (1) of a degree greater than or equal to two. For this, we need the following preliminary result, which describes the Darboux polynomials and their cofactors of system (1) restricted to each plane H j , j = 1, 2, 3, defined in (2).
It follows from Propositions 6-8 that i.e., l 2 = j 2 = m 2 . Furthermore, we have We note that the right-hand side of (27) is always nonnegative while the quantity −n − l 2 is always nonpositive. It hence follows that these expressions must be zero, n = l 2 = 0, in contradiction with n ≥ 2. The proposition is proved.
Proof of Theorem 7. If f has degree one, then the proof follows directly from Proposition 4. We now assume that system (1) has an irreducible Darboux polynomial of a degree at least two with the cofactor K = 0 given as in Lemma 2. From Lemma 3, we can then conclude that f is a homogeneous irreducible Darboux polynomial of degree at least two and the cofactor K = 0. We then obtain a contradiction from Proposition 5. Therefore, all Darboux polynomials with a cofactor K = 0 given as in Lemma 2 must come from Darboux polynomials of degree one, i.e., H 1 , H 2 , or H 3 . Furthermore, it follows from Theorem 1 that all Darboux polynomials with a zero cofactor, i.e., all polynomial first integrals, must be constants. Hence, applying Lemma 5, we obtain the proof of Theorem 7.
Proof of Theorem 3. By Theorem 7, it follows that every Darboux polynomial of system (1) has the form f = cH n1 where c is some constant and n 1 , n 2 , and n 3 are nonnegative integers. Furthermore, the cofactor of f is given in (22). From Proposition 1 and Theorem 1, we find that the existence of a nonconstant rational first integral implies the existence of two coprime Darboux polynomials with the same nonzero cofactor. Hence, the first integral must have the form 3 ) with at least one nonzero m i and n i , and the cofactors of R and S must be equal. According to (22), the equality of the cofactors of R and S then implies that Hence, because the α i are real, we obtain m i = n i for i = 1, 2, 3, which contradicts the fact that R and S are coprime. The theorem is thus proved.

Proof of Theorem 4
We recall that the equation defining the exponential factor F = e h/g with the cofactor L for system (1) isẋ where we simplify the common factor F and According to Propositions 2 and 3 and Theorems 1 and 7, if system (1) where L is given in (29). Taking x 1 = x 2 = x 3 = 0 in (30), we obtain b 0 = 0. Setting x 1 = x 2 = 0 in (30), we now obtain This equation implies that b 3 = 0. Analogously, setting x 1 = x 3 = 0 in (30), we obtain b 2 = 0, and setting x 2 = x 3 = 0 in (30), we obtain b 1 = 0. Therefore, L = 0, and from (30), we find that h is a polynomial first integral of system (1), which contradicts Theorem 1.