DARBOUX INTEGRABILITY OF 2 – DIMENSIONAL HAMILTONIAN SYSTEMS WITH HOMOGENOUS POTENTIALS OF DEGREE 3

We provide a characterization of all Hamiltonian systems of the form H = (p1 + p 2 2)/2 + V (q1, q2) where V is a homogenous polynomial of degree 3 which are completely integrable with Darboux first integrals.


Introduction and statement of the main results
We consider R 4 as a symplectic linear space with canonical variables q = (q 1 , q 2 ) and p = (p 1 , p 2 ), with q i called the coordinates and p i called the momenta.We want to study the Hamiltonian systems with Hamilton's function of the form where V (q) = V (q 1 , q 2 ) is a homogeneous polynomial of degree 3, i.e. we will study the Hamiltonian systems (1) qi = p i , ṗi = − ∂V ∂q i , i = 1, 2.
During the last century these systems have been actively investigated, with special attention to these systems where V has degree at most 5 and which have a second polynomial first integral with degree at most 4 in the variables p 1 and p 2 (see for instance [1,2,10,11,18]).When the homogeneous potential has degree three we want to mention the work [12] where he studied the polynomial integrability assuming that the additional first integral is a polynomial of order at most four in the momenta.
We start by recalling some definitions.Let A = A(p, q) and B = B(p, q) be two functions.Their Poisson bracket {A, B} is defined as ) .
We say that two functions A and B are in involution if {A, B} = 0. We say that a non-constant function F = F (q, p) is a first integral for the Hamiltonian system (1) if it commutes with the Hamiltonian function H, i.e., {H, F } = 0. We say that the Hamiltonian system (1) is completely integrable if it has 2 functionally independent first integrals which are in involution.One first integral will always be the Hamiltonian H.We say that two functions H and F are independent if their gradients are linearly independent at all points R 4 except perhaps in a zero Lebesgue set.
A polynomial first integral f = f (x, y, p x , p y ) of system (1) is a first integral which is a polynomial, and a rational first integral of system (1) is a first integral which is a rational function.To study the existence of rational first integrals we will use the well-known Darboux theory of integrability.The Darboux theory of integrability in dimension 4 is based on the existence of invariant algebraic hypersurfaces (or Darboux polynomials).For more details see [4,5] and [13].This theory is one of the best theories for studying the existence of first integrals for the polynomial differential systems.
A Darboux polynomial of system ( 1) is a polynomial f ∈ C[p 1 , p 2 , q 1 , q 2 ] \ C such that (2) for some polynomial K called the cofactor of f and with degree at most one.
Note that f = 0 is an invariant algebraic hypersurface for the flow of system (1), and a polynomial first integral if it is a Darboux polynomial with zero cofactor.We recall that if f ̸ ∈ R[p 1 , p 2 , q 1 , q 2 ] \ R is a Darboux polynomial then there exists another Darboux polynomial f (the conjugate of f ) with cofactor K (the conjugate of K).
for some polynomial L = L(p 1 , p 2 , q 1 , q 2 ) called the cofactor of F and with degree at most one.We recall that if F ̸ ∈ R[p 1 , p 2 , q 1 , q 2 ] \ R is an exponential factor then there exists another exponential factor F (the conjugate of F ) with cofactor L (the conjugate of L).
A Darboux first integral G of system (1) is a first integral of the form , where f 1 , . . ., f p are Darboux polynomials and F 1 , . . ., F q are exponential factors and λ j , µ k ∈ C for j = 1, . . ., p and k = 1, . . ., q.Note that a Darboux first integral is always a real function due to the fact that if there are complex Darboux polynomials or complex exponential factors always also appear their conjugates.
We consider the Hamiltonian systems (1) with a homogeneous potential of degree 3 of the form (4) Let PO 2 (C) denote the group of 2×2 complex matrices A such that AA T = α Id, where Id is the identity matrix and α ∈ C \ {0}.The potentials V 1 (q) and V 2 (q) are equivalent if there exists a matrix Therefore we divide all potentials into equivalent classes.In what follows a potential means a class of equivalent potentials in the above sense.This definition of equivalent potentials is motivated by the following simple observation (for a proof see [12]).Let V 1 and V 2 be two equivalent potentials, if the Hamiltonian system with Hamiltonian (5) and potential A very relevant result related with the integrability of Hamiltonian systems with homogeneous potential was given by Morales and Ramis (see page 100 of [17] and the references therein), with a method that gives a necessary condition for the existence of an additional meromorphic first integral.Using these results, Maciejewski and Przybylska [15,16] proved the following result.
Theorem 1.The Hamiltonian systems (1) with homogeneous potential of degree 3 is completely integrable with meromorphic first integrals if and only if V is one of the potentials Unfortunately Theorem 1 is not completely correct because the potential has an additional polynomial first integral given in Table 1, which of course is a meromorphic first integral.We find this new integrable case in the proof of Proposition 7 studying the exponential factors of system (1).Our main result is the following one.
Theorem 2. The Hamiltonian systems (1) with homogeneous potential of degree 3 is completely integrable with Darboux first integrals if and only if V is one of the potentials given in Table 1 with V 0 , V 1 , . . ., V ± 8 given in (6) and V 9 given in (7).In Table 1 we also provide the additional independent polynomial first integral for each potential.
We recall that not all meromorphic functions are Darboux functions, and that not all Darboux functions are meromorphic functions.This shows that Theorems 1 and 2 are independent.
In section 2 we provide a brief introduction to the quasi-homogenous polynomial differential systems that will allow to prove Theorem 2. In section 3 we give the proof of Theorem 2. We recall that since we cannot find in the literature the explicit expressions of all these polynomial first integrals for the nine potentials of Table

Potential
First integral Table 1.All nonequivalent integrable homogeneous polynomial potentials of degree 3 with their corresponding additional first integral.Here i = √ −1.
1, we shall provide them here, we have used for their computation the theory of integrability for the quasi-homogeneous polynomial differential systems.

Quasi-homogeneous polynomial differential systems
In this subsection we summarize some basic results on the analytic and polynomial integrability of the polynomial differential systems of the form ( 8) As usual N, R and C will denote the sets of positive integers, real and complex numbers, respectively; and Here t can be real or complex.The polynomial differential system (8 (8), and d the weight degree with respect to the weight exponent s.In the particular case that s = (1, • • • , 1) system ( 8) is the classical homogeneous polynomial differential system of degree d.
Note that if the polynomial differential system ( 8) is quasi-homogeneous with weight exponent s and weight degree d > 1, then the system is invariant under the change of variables Recently the integrability of quasi-homogeneous polynomial differential systems have been investigated by several authors.Probably the best results have been provided by Yoshida [20,21,22], Furta [7] and Goriely [9], see also Tsygvintsev [19] and Llibre and Zhang [13].
A non-constant function constant for all values of t for which the solution (x The function F (x 1 , . . ., x n ) is quasi-homogeneous of weight degree m with respect to the weight exponent s if it satisfies Given an analytic function F we can split it in the form F = ∑ i F i , where F i is a quasi-homogeneous polynomial of weight degree i with respect to the weight exponent s; i.e.F i (α The following result is well known, see for instance Proposition 1 of [13]. Proposition 3. Let F be an analytic function and let F = ∑ i F i be its decomposition into weight-homogeneous polynomials of weight degree i with respect to the weight exponent s.Then F is an analytic first integral of the weight-homogeneous polynomial differential system (8) with weight exponent s if and only if each weight homogeneous part F i is a first integral of system (8) for all i.
Suppose that system ( 8) is a quasi-homogeneous polynomial differential system of weight degree d with respect to the weight exponent s.Then we define w = s/(d − 1).The interest for the quasi-homogeneous polynomial differential systems is based in the existence of the particular solutions of the form where the coefficients c = (c 1 , . . ., c n ) ∈ C n \ {0} are given by the algebraic system of equations For a given (w 1 , . . ., w n ) there may exist different c's, called the balances.
For each balance c we introduce a matrix where as usual DP(c) denotes the differential of P evaluated at c, and diag(w 1 , . . ., w n ) denotes the matrix whose diagonal is equal to (w 1 , . . ., w n ) and zeros in the rest.The eigenvalues of K(c) are called the Kowalevsky exponents of the balance c.Sophia Kowalevskaya was the first to introduce the matrix K to compute the Laurent series solutions of the rigid body motion.It can be shown that there always exists a Kowalevsky exponent equal to −1 related to the arbitrariness of the origin of the parametrization of the solution by the time.The eigenvector associated to the eigenvalue −1 is wc = (w 1 c 1 , . . ., w n c n ), for more details see [20] or [7].
A non-constant function constant for all values of t for which the solution (x Probably the best results in order to know if a weighted homogeneous polynomial of weight degree m with respect to the weight exponent s is a first integral of a quasi-homogeneous polynomial differential system (8) with weight degree d with respect to the weight exponent s is essentially due to Yoshida [20], and are the following two theorems.Theorem 4. Let F (x 1 , . . ., x n ) be a weighted homogeneous polynomial first integral of weight degree m with respect to the weight exponent s of the quasi-homogeneous polynomial differential system (8) with weight degree d with respect to the weight exponent s.Suppose the gradient of F evaluated at a balance c is finite and not identically zero.Then m/(d − 1) is a Kowalevsky exponent of the balance c.
Let r be a positive integer such that 1 < r < n.A set of functions is r for all the x ∈ C n , except perhaps in a zero Lebesgue subset of C n .
Theorem 5. Let r be a positive integer such that 1 < r < n, and let F k (x 1 , . . ., x n ) for k = 1, . . ., r be weighted homogeneous polynomial first integrals of weight degree m with respect to the weight exponent s of the quasi-homogeneous polynomial differential system (8) with weight degree d with respect to the weight exponent s.
Suppose that the gradients of F k for k = 1, . . ., r evaluated at a balance c are finite, not identically zero and independent.Then m/(d − 1) is a Kowalevsky exponent of the balance c with multiplicity at least r.

Proof of Theorem 2
We divide the proof of Theorem 2 into different subsections.In the first subsection we study the polynomial first integrals.
3.1.The polynomial first integrals.Now we shall compute the polynomial first integrals of the Hamiltonian systems with the homogeneous polynomial potentials given in Table 1.
The Hamiltonian system associated to the Hamiltonian ( 5) is ) is a homogeneous polynomial of degree 3, then this Hamiltonian system is a quasi-homogeneous polynomial differential system (8) with x = (q 1 , q 2 , p 1 , p 2 ) and weight degree d = 2 with respect to the weight exponent s = (s 1 , s 2 , s 3 , s 4 ) = (2, 2, 3, 3).Hence we can apply the theory for finding polynomial first integrals to our Hamiltonian systems (9) described in the subsection 2.
The potential V 0 of Table 1 is Unfortunately if we compute for the Hamiltonian system (9) with the potential V = V 0 its balances we see that this differential system has no balances.So the theory of quasi-homogeneous polynomial differential systems for finding polynomial first integrals cannot be applied to this Hamiltonian system.But it is easy to check that is a polynomial first integral for this Hamiltonian system.The potential V 1 of Table 1 is The Hamiltonian system (9) with the potential ) .
The Kowalevsky exponents of this balance are −1, −1, 6 and 6.By Theorem 4 if H 2 (q 1 , q 2 , p 1 , p 2 ) is a weighted homogeneous polynomial first integral of weight degree m with respect to the weight exponent (2, 2, 3, 3) of the quasi-homogeneous polynomial differential system (9) with weight degree d = 2 with respect to the weight exponent (2, 2, 3, 3), whose gradient evaluated at the balance c is finite and not identically zero, then m/(d − 1) = m is a Kowalevsky exponent.So if a such polynomial first integral exists its weighted degree is 6.Looking for these kind of polynomial first integral we obtain that , is a polynomial first integral of the Hamiltonian system (9) The potential V 2 of Table 1 is The Hamiltonian system (9) with the potential V = V 2 has the balance ) .
The Kowalevsky exponents of this balance are −1, −1, 6 and 6.Again by Theorem 4, if the corresponding Hamiltonian system has a polynomial first integral under the assumptions of that theorem its weighted degree must be 6.Looking for these kind of polynomial first integral we obtain that The potential V 3 of Table 1 is The Hamiltonian system (9) with the potential V = V 3 has no balances.But it is easy to check that H 3 = p 1 + ip 2 , is a polynomial first integral of the Hamiltonian system (9) with V = V 3 .
The potential V 4 of Table 1 is 1 .The Hamiltonian system (9) with the potential V = V 4 has the balance c = (−2, 0, 4, 0).The Kowalevsky exponents of this balance are −1, 2, 3 and 6.Again by Theorem 4, if the corresponding Hamiltonian system has a polynomial first integral under the assumptions of that theorem its weighted degree must be 2, 3 or 6.Looking for these kind of polynomial first integral we obtain that H 4 = p 2 , or H4 = p 2 1 + 2q 3  1 , are polynomial first integrals of the Hamiltonian system (9) with V = V 4 .
The potential V 5 of Table 1 is ).The Hamiltonian system (9) with the potential V = V 5 has the following three balances if b ̸ = 0 and only one b = 0: ) .
The Kowalevsky exponents of the first two balances are −1, 2, 3 and 6, and of the third balance are −1, −1, 6 and 6.Again by Theorem 4, if the corresponding Hamiltonian system has a polynomial first integral under the assumptions of that theorem its weighted degree must be 2, 3 or 6.Looking for these kind of polynomial first integral we obtain that , are polynomial first integrals of the Hamiltonian system (9) with V = V 5 .
The potential V 6 of Table 1 is q 2 1 q 2 + q 3 2 .The Hamiltonian system (9) with the potential V = V 6 has the following three balances The Kowalevsky exponents of the first balance is −1, 1, 4 and 6, and of the last two balances are −3, −1, 6 and 8. Again by Theorem 4, if the corresponding Hamiltonian system has a polynomial first integral under the assumptions of that theorem its weighted degree must be 4, 6 or 8. Looking for these kind of polynomial first integral we obtain that ), is a polynomial first integral of the Hamiltonian system (9) with V = V 6 .
The potential V 7 of Table 1 is 2 .
The Hamiltonian system (9) with the potential V = V 7 has the following three balances The Kowalevsky exponents of the first balance is −1, 3/2, 7/6 and 6, and of the last two balances are −7, −1, 6 and 12. Again by Theorem 4, if the corresponding Hamiltonian system has a polynomial first integral under the assumptions of that theorem its weighted degree must be 6 or 12. Looking for these kind of polynomial first integral we obtain that ), is a polynomial first integral of the Hamiltonian system ( 9) with V = V 7 .
The potential V ± 8 of Table 1 is The Hamiltonian system (9) with the potential V = V ± 8 has the following three balances The Kowalevsky exponents of the first balance are −1, 1, 4 and 6, of the second are −2, −1, 6 and 7, and of the third are −7, −1, 6 and 12. Again by Theorem 4, if the corresponding Hamiltonian system has a polynomial first integral under the assumptions of that theorem its weighted degree must be 1, 4, 6 or 12. Looking for these kind of polynomial first integral we obtain that , is a polynomial first integral of the Hamiltonian system (9) with V = V ± 8 .The potential V 9 of Table 1 is The Hamiltonian system (9) with the potential V = V 8 has the following three balances c 1 = (0, −12, 0, 24), ) , ) .
The Kowalevsky exponents of the first balance are −1, −1, 6 and 6, of the second and third balances are −1, 2, 3 and 6.Again by Theorem 4, if the corresponding Hamiltonian system has a polynomial first integral under the assumptions of that theorem its weighted degree must be 2, 3 or 6.Looking for these kind of polynomial first integral we obtain that is a polynomial first integral of the Hamiltonian system (9) with V = V 9 .
3.2.The Darboux polynomials with nonzero cofactor.Now we shall compute the Darboux polynomials of system (1) with nonzero cofactor.It was proved in [15] that there exists an affine change of variables in such a way that the Hamiltonian system (1) with homogeneous potential (4) can be written in the new variables as reduced as a Hamiltonian system (1) wtith homogeneous potential From now on, we will work with the homogenous potential (10).Proposition 6.The Hamiltonian system (1) with homogeneous potential (10) has no Darboux polynomials with nonzero cofactor.
Since f 1 must be a polynomial and α 0 ̸ = 0 we must have that K 1 = 0 and thus f 1 = 0. Proceeding inductively we get that f j = 0 for j = 2, 3, . . ., n in contradiction with the fact that f n ̸ = 0.This concludes the proof of the theorem.
Now we shall study the exponential factors of system (1).In view of subsections 3.1 and 3.2, and Theorem 1, the unique homogeneous potentials of the form (10) which admit an additional Darboux polynomial are the ones given in Table 1.Therefore, we will only study the systems with potentials that do not belong to Table 1.For these systems we know that the unique Darboux polynomial is the Hamiltonian H with cofactor zero.

The exponential factors.
In this subsection we will prove the following result.
Proposition 7. The unique exponential factors for the Hamiltonian system (1) with homogeneous potential (10) that do not belong to Table 1 are e P (H)/Q(H) (where P and Q are polynomials in the variable H), e q 1 , e q 2 and linear combinations of all the exponents in the exponential.
To prove Proposition 7 we will use the following two known results whose proof and geometrical meaning is given in [3,14].Proposition 8.The following statements hold.
(a) If E = exp(g 0 /g 1 ) is an exponential factor for the polynomial system (1) and g 1 is not a constant polynomial, then g 1 = 0 is an invariant algebraic hypersurface.(b) Eventually e g 0 can be an exponential factor, coming from the multiplicity of the infinite invariant hyperplane.
The following result given in [14] characterizes the algebraic multiplicity of an invariant algebraic hypersurface using the number of exponential factors of system (1) associated with the invariant algebraic hypersurface.Theorem 9. Given an irreducible invariant algebraic hypersurface g 1 = 0 of degree m of system (1), it has algebraic multiplicity k if and only if the vector field associated to system (1) has k − 1 exponential factors of the form exp(g 0,i /g i 1 ), where g 0,i is a polynomial of degree at most im and (g 0,i , g 1 ) = 1 for i = 1, . . ., k − 1.
In view of Theorem 9 if we prove that e g 0 /g 1 is not an exponential factor with degree g 0 ≤ degree g 1 , there are no exponential factors associated to the invariant algebraic curve g 1 = 0.
Proof of Proposition 7. Note that H is an irreducible polynomial in C[q 1 , q 2 , p 1 , p 2 ].Indeed, H = H 1 H 2 , and consequently H 1 and H 2 will be Darboux polynomials.By Proposition 6, H 1 and H 2 will be two polynomial first integrals of system (1) of degree at most 2, and such first integrals do not exist.
System (1) has the irreducible Darboux polynomial H. Then in view of Proposition 8 it can have an exponential factor of the form: either E = exp(g) with g ∈ C[p 1 , p 2 , q 1 , q 2 ]\C, or E = exp(g/H m ) with m ≥ 1 and such that g ∈ C[p 1 , p 2 , q 1 , q 2 ] is coprime with H.
We first prove that system (1) has no exponential factors of the form E = exp(g) with g ∈ C[p 1 , p 2 , q 1 , q 2 ] \ C. In this case, g satisfies where we have simplified the common factor E = exp(g) and where Evaluating ( 16) on p 1 = p 2 = q 1 = q 2 = 0 we get β 0 = 0. We write h as h = ∑ n j=0 h j (p 1 , p 2 , q 1 , q 2 ) where each h j is a homogeneous polynomial of degree j in each variables p 1 , p 2 , q 1 and q 2 .Now computing the terms of degree one in (16) we get Clearly, setting p 1 = p 2 = 0 in the above equality we get β 3 = β 4 = 0.In short, h satisfies In other words, h is either a constant or a polynomial first integral of system (1).By hypothesis we have that h = h(H) where h can be constant.Hence, in this case g = h(H) + β 1 q 1 + β 2 q 2 , β 1 , β 2 ∈ C.
Assume now that system (1) has an exponential factor of the form E = exp(g/H m ) with m ≥ 1 such that H and g ∈ C[p 1 , p 2 , q 1 , q 2 ] are coprime.In view of Theorem 9 we can assume that m = 1 and that g has degree at most three (note that here g 1 = H has degree three).We write g as a polynomial of degree three in the variables p 1 , p 2 , q 1 , q 2 and since e g/H must be an exponential factor g must satisfy p 1 ∂g ∂q 1 + p 2 ∂g ∂q 2 − (aq 2 1 + q 1 q 2 ) ∂g ∂p 1 − (q 2 1 /2 + bq 2 2 ) ∂g ∂p 2 = (β 0 + β 1 p 1 + β 2 p 2 + β 3 q 1 + β 4 q 4 ) ( p 2 1 + p 2 2 2 + a 3 2 ) . ( Computing the solutions of ( 17) using an algebraic manipulator such as mathematica and recalling that by assumptions we have b ̸ = 0 we get that β i = 0 for i = 0, . . ., 4, i.e.L = 0 and g is either const H, or g = const/H, or g = H 9 .So, Proposition 7 is proved.
Now we have all the ingredients to prove Theorem 2 3.4.Proof of Theorem 2. In order to proof Theorem 2 we need the following result whose proof is given in [6].
Theorem 10.Suppose that system (1) admits p Darboux polynomials and with cofactors K i and q exponential factors F j with cofactors L j .Then there exists λ j , µ j ∈ C not all zero such that if and only if the function G given in (3) (called of Darboux type) is a first integral of system (1).
In view of Theorem 10 to characterize the Darboux first integrals we need to compute the Darboux polynomials and the exponential factors.Then, under our assumptions, using Propositions 6 and 7 if G is a Darboux first integral of system (1) it must be of the form (3), i.e., G = P 1 (H)e λP (H)/Q(H)+µ1q1+µ2q2 for some P 1 , P, Q polynomials and λ, µ 1 , µ 2 ∈ C.Moreover, the cofactors must satisfy µ 1 p 1 + µ 2 p 2 = 0.
Solving this system we have µ 1 = µ 2 = 0 which shows that G = P 1 (H)e λP (H)/Q(H) only depends on H.This concludes the proof of Theorem 2.