Chaos, Solitons and Fractals

We characterize, using normal forms of quasi-homogeneous expansions, the analytic vector ﬁelds at nilpotent singular point having an algebraic ﬁrst integral over the ring C [[ x , y ]] . As a consequence, we provide a link between the algebraic integrability problem and the existence of a formal inverse integrating factor which is null at the singular point.


Introduction
We consider an autonomous system ˙ x = F (x ) where F = (P, Q ) T is a planar vector field defined in a domain U ⊂ C 2 whose components P and Q are analytic in a neighborhood of the origin and coprimes.Let ⊂ U be with U \ open, a set of integral curves of the system, we say that a continuous function H : U \ → C is a first integral of F if H is constant on each solution of system ˙ x = F (x ) contained in U \ , and H is non-constant on any open subset of U \ .If is a empty set, H is a strong first integral; otherwise, H is a weak first integral.Thus, the first integrals computed by Darboux [15] in 1878 for polynomial systems are weak first integrals, in general.
Here, we study the existence of an algebraic first integral, i.e. a first integral which is solution of a polynomial with coefficients over C [[ x, y ]] (ring of the formal series in x, y with coefficients on C ).As we will see below ( Proposition 4 , the existence of an algebraic first integral is equivalent to the existence of a first integral that is a ratio of two coprime formal functions. A function H = f g with f and g analytic or formal functions, is a first integral of F if H is a weak first integral on the open set g = { (x, y ) ∈ U : g(x, y ) = 0 } .Obviously, if H is a first integral, then Lie derivative of H by F is zero in g , i.e.F (H) = 0 in g , where F := P ∂x + Q ∂y denotes the differential operator associated to the vector field F .It will cause no confusion if we use the same letter to designate the vector field and its associated differential operator.
If the open U does not have any singular point, from flows box's theorem [12, Cauchy-Arnold Theorem] , the vector field is analytically integrable, i.e. there always exists an analytic first integral of Goriely [22] and Nowicki [28] .Shi [32] studies the nonexistence of rational first integral for semiquasihomogeneous systems.
Two analytic differential systems are called orbitally equivalent if there exists a transformation of the phase space which takes complex phase curves of the first to those of the second one or equivalently, there exist a formal near-identity change of variables and a formal re-parameterization of the time-variable that transform one into the other one.
By [17,23] , the nondegenerate center-focus type and resonant saddles singular points, have an analytic local first integral if, and only if, they are orbitally linearizable.
In this paper, we deal with analytic differential systems ˙ x = y + X (x, y ) , ˙ y = Y (x, y ) , (1) with X and Y analytic functions without constants and linear terms.Thus, the linear part valued at the origin is non-zero and the eigenvalues of the linear part of the vector field are zero at that point, i.e. the origin is a nilpotent singular point.
The analytic integrability problem around a nilpotent singularity is already solved.Algaba et al. [1] study the vector fields whose leader quasi-homogeneous term is Hamiltonian, see [1, Theorem 3.19] .Recently, Algaba et al. [3] have solved the analytic integrability problem around a nilpotent singularity of a planar vector field under generic conditions (the origin of the leader quasi-homogeneous vector field is an isolated singular point), see [3,Theorem 1.2] .Later, Algaba et al. [2] have solved the remaining case, completing the analytic integrability problem for such singularity, see [2,Theorem 1.4] .In all the cases, the vector field has an analytic local first integral if, and only if, it is orbitally quasi-homogenizable (orbitally equivalent to its quasihomogeneous leader term).
Here, we study the nilpotent vector fields which have an algebraic first integral on C [[ x, y ]] .Because the eigenvalues are zero, Poincaré-Dulac normal form theory is not applicable and so we need an appropriate normal form for the study of this problem.
We introduce some notation and concepts to present the main results.Given t = (t 1 , t 2 ) with t 1 and t 2 natural numbers without common factors, a scalar function f of two variables is a quasihomogeneous function of type or weight exponent t and degree The vector space of quasi-homogeneous polynomials of type t and degree j is denoted by P t j .A vector field .We denote the vector space of the quasi-homogeneous polynomial vector fields of type t and degree j by Q t j .An analytic vector field can be expanded into quasihomogeneous terms of type t of successive degrees.Thus, the vector field F can be written in the form for some integer r, where F j = (P j+ t 1 , Q j+ t 2 ) T ∈ Q t j and F r ≡ 0 .As mentioned before, the nilpotent vector fields are analytically integrable if, and only if, they are orbitally equivalent to a polynomially integrable quasi-homogeneous vector fields, see [1][2][3] .In this paper, we solve the remaining case, completing the algebraic integrability problem for such singularity; that is, we address our study when the vector field has an algebraic first integral but it does not have any analytic first integral and we obtain the following result.
Theorem 1. (Algebraic and non-analytic integrability of nilpotent vector fields).We assume that F is an analytic vector field whose origin is a nilpotent singularity and F is not analytically integrable.The vector field F has an algebraic first integral if, and only if, after a polynomial change of variables (if necessary), is formally orbitally equivalent to with n a natural number and d a rational number with | d| > 1 , i.e. a quasi-homogeneous vector field of weight t = (1 , n + 1) and degree n.
Next result allows to extend the results of [17,23,25] on algebraic integrability for vector fields whose origin is a nondegenerate or elementary singular point to the nilpotent vector fields.

Theorem 2. (Algebraic integrability of nilpotent vector fields).
Let F be an analytic vector field with a nilpotent singularity at the origin.The vector field F has an algebraic first integral if, and only if, it is orbitally equivalent to a rationally integrable quasi-homogeneous vector field.
We also provide a characterization of algebraic integrability (analytic or not) through the existence of an inverse integrating factor, i.e. a formal function V, zero at origin, such that The following result is a tool we will use to obtain necessary and sufficient conditions of algebraical integrability.The condition on the expression of the lowest-degree term of the inverse integrating factor can not be removed.We remark that there exist nilpotent vector fields which are not algebraically integrable and nevertheless have an algebraic (and formal) inverse integrating factor.For example, the polynomial (x 4 − 2 y 2 ) 2 is an inverse integrating factor of the vector field (y The rest of the paper is organized as follows: In Section 2 , we provide a normal pre-form of a nilpotent vector field with an algebraic first integral, Proposition 5 .Here we also include some properties of the invariant curves of this normal pre-form.Section 3.1 contains a normal form of an analytic planar vector field using quasi-homogeneous expansion, see Theorem 8 .We include this section trying to do the paper self-contained.The results shown and their proofs, we can find them in [3] .The results of subsection 3.1 are new, we give a orbitally equivalent normal form for the vector fields whose quasi-homogeneous leader term is rationally integrable but no polynomially integrable, see Theorems 9 and 10 .Section 4 is the proof of Theorem 1, 2 and 3 and Section 5 contains some technical results we use for proving the above results.Last on, in Section 6 , we solve our problem for a polynomial family by providing the algebraic first integrals.

Necessary conditions of algebraical and non-analytical integrability of nilpotent vector fields
We first give necessary conditions of algebraic integrability.We recall that an invariant curve at the origin of a vector field F is a formal curve C(x, y ) = 0 , null at the origin, satisfying F (C) = KC with K formal function, named cofactor of C.

Proposition 4. Consider a non-analytically integrable vector field. If it has an algebraic first integral at the origin then it has a first integral that is ratio of two coprime formal functions. Moreover, it has two formal invariant curves at the origin with the same cofactor.
Proof.We assume that the analytic vector field has an algebraic first integral and it is not analytically integrable.From [30, Propositions 1 and 2] , it also admits a first integral V of the spe- This property also is satisfied in W 1 since it is enough to consider the first integral . Therefore, the property is true on C 2 since W 1 ∩ W 2 = { 0 } .Thus, W 1 and W 2 are formal invariant curves with the same cofactor.
We give a normal pre-form of nilpotent vector fields with an algebraic first integral.This result allows to obtain necessary conditions of algebraic integrability.

Proposition 5. We assume that F is an analytic vector field whose origin is a nilpotent isolated singularity and has an algebraic first integral. There exists a polynomial change of variables φ such that
where F r is one of the following vector fields: Moreover, in this case, F is analytically integrable.
Moreover, in this case, F is analytically integrable.
Proof.A normal pre-form of analytically integrable nilpotent systems can be found in [3] .We study the algebraic and non-analytic integrability.
According to the kind of the Newton diagram of a vector field and therefore depending on the type and degree of the quasihomogeneous leader term, a vector field F whose origin is a nilpotent singularity can be transformed, by means of a polynomial and only one, of the following quasi-homogeneous vector fields: that is, the components of the vector field are not coprimes (the origin is a not isolated singular point) and it is always analytically integrable by the flow box's theorem, see [12] .
Therefore, by Proposition 4 , φ * F does not have any algebraic and non-analytic first integral (case A) ).The analytic integrability of the vector field * F has been studied in [1] .
• F r = y∂ x + bx n y∂ y , b = 0 , i.e. t = (1 , n + 1) and r = n.The vector field φ * F has, at most, two irreducible invariant curves at origin of the form (n to degree n are 0 and bx n , respectively.Therefore, these curves do not have the same cofactor, by Proposition 4 , φ * F does not have any algebraic and non-analytic first integral (case B) ).As a consequence, if there exists an algebraic first integral then the vector field is analytically integrable.
If c = −1 , there exists an unique irreducible analytic invariant curve at origin starting by y 2 + x 2 n +2 and therefore φ * F does not have any algebraic first integral.If c = 0 , the vector field * F is not analytically integrable since its Hamiltonian part has multiple factors, see [7,Theorem 3.2] .
In this case, the invariant curves, of existing, are invariant curves of the form y + • • • .We assume that the vector field has two invariant curves f 1 and f 2 with the same cofactor.This fact arrives to a contradiction since f 1 − f 2 is also an invariant curve of F r and it does not star with y .Thus, the vector field * F does not have any algebraic first integral.
Otherwise, c = 1 .We distinguish two cases: if with n a natural number and d a rational number with | d| > 1 .
The existence of invariant curves of system (2) is provided by Algaba et al. [6] .Proposition 6. ( [6 , Theorem 19]) Let F be the vector field (2) .It holds: (i) if d ∈ Q and d > 1 , F has a unique irreducible analytic invariant curve at the origin starting with y − x n +1 .
Moreover, it has an irreducible analytic invariant curve at the origin starting with y

has a unique irreducible analytic invariant curve at the origin starting with y
Moreover, it has an irreducible analytic invariant curve at the origin starting with y We emphasize that if C is a formal or analytic invariant curve at the origin, then for any unit function with K cofactor of C and u (0 ) = 1 .As a consequence of [6, Theorem 5] , at least one of the formal irreducible invariant curves of (2) starting with y − x n +1 (or y + x n +1 ) is analytic.
The following result provides a canonical expression of the invariant curves starting with y − x n +1 or y + x n +1 .
We consider the sum We know that Cu = (y an invariant curve of F for any unit function u .

The term of degree
and we choose u j such that −(y −

Normal form of quasi-homogeneous expansions under orbital equivalence
We include this section trying to do the paper self-contained.In the first part of this section, we provide a normal form of an analytic planar vector field using quasi-homogeneous expansion, Theorem 8 .Its proof we can find it in [3] .
We consider a general vector field The key in the problem of obtaining a normal form of the vector field F is to analyze the effect that a near-identity transformation x = y + P j (y ) and a reparametrization of the time by dt dT = 1 + τ j (x ) has on the system ˙ x = F (x ) , where P j ∈ Q t j and τ j ∈ P t j with j ≥ 1 .The quasi-homogeneous terms of the transformed system y = d y dT = G (y ) agree with the original ones up to degree r + j − 1 and for the degree r + j it holds where we have introduced the homological operator under orbital equivalence and [ P j , F r ] := D P j F r − D F r P j is the Lie bracket of P j and F r .Following the ideas of the conventional normal form theory, by means of a sequence of time-reparameterizations and near-identity transformations, the system ˙ x = F (x ) can be formally reduced to normal form under orbital equivalence, i.e. the system can be transformed into where Cor (L r+ j ) is any complementary subspace to the range of the homological operator L r+ j .We note that such space is not unique, in general.
Here, we give an expression of L r+ j we will use to obtain an expression of Cor (L r+ j ) , or equivalently, we provide a normal form of the vector field.
A main role plays the linear operator r+ j , Lie-derivative operator of F r , i.e. r+ j : P t j −→ P t r+ j (3) The operator L r+ j restricted to Q t j × Cor ( j ) and the operator L r+ j have the same range.In this way, by keeping the notation, we consider to L r+ j restricted to Q t j × Cor ( j ) as the homological operator under orbital equivalence.
The following subspaces of Q t j will be useful in the study of the homological operator under orbital equivalence.Notice that P t j−r = { 0 } if j < r and , for all j ∈ N .This decomposition of the quasi-homogeneous vector fields allows to define the corresponding projectors c , d and f .Also, we can identify j and λ ∈ P t j−r such that P j = X δ + ηD 0 + λF r , we denote c P j = P c j := Proj C t j P j = X δ , d P j = P d j := Proj D t j P j = ηD 0 and f P j = P f j := Proj F t j P j = λF r .
With this notation the homological operator under equivalence can be written as where c Under certain conditions on c r+ j+ | t | , we can give an orbitally equivalent normal form of an analytic vector field.Proof.By using the decomposition in direct summa of initial and final spaces of the operator L r+ j and writting F t r+ j = Range j F r Cor j F r , we obtain the following matrix diagonal by blocks of the operator L r+ j :

Theorem 8. Given
From hypothesis Ker c r+ j+ | t | = { 0 } , we can deduce that the upper left block of the matrix has maximum range.Thus, we can choose the following subspace complementary to range of L r+ j , and, therefore, the result follows.
Remark 1. Suppose that there is a j 0 such that Ker c r+ j 0 + | t | is not a trivial set.We consider the linear operator L r+ j 0 : X the second component of the operator L r+ j 0 .
If L r+ j 0 has full range, the quasi-homogeneous term of degree r + j 0 of the normal form is G r+ j 0 = X δ with δ ∈ Cor c r+ j 0 + | t | .

Normal form under orbital equivalence of nilpotent systems whose first quasi-homogmeous component are algebraically but non-analytically integrable
We give an orbitally equivalent normal form of (2) with d rational number and | d| > 1 , nilpotent vector field whose lowest-degree quasi-homogeneous term is rationally and nonpolynomially integrable.We distinguish two cases: , for all k natural number.The vector field ( 2) is orbitally equivalent to G = Proof.The linear operator n + j given by (3) is the operator ˆ n + j for C = 0 given by (10) .Thus, applying Lemma 12 for C = 0 , we have that a complementary subspace to the range of the linear operator n + j , with j = i 0 + l 0 (n + 1) , 0 ≤ i 0 < n + 1 , is Cor ( n + j ) = { 0 } if i 0 = 0 , and Cor ( n + j ) = x n + j if i 0 > 0 .
, for some k 0 natural number.

The vector field (2) is orbitally equivalent to
Proof.Reasoning as before, we have that a complementary subspace to the Range of the linear operator j+ n given by (3) , with On the other hand, if d > 0 (for d < 0 the proof is analogous), the linear operator c k 0 +2(n +1) given by ( 5) is the operator ˆ c k 0 +2(n +1) for C = − 2(n +1) k 0 +2(n +1) d given by (11) .By hypothesis, we have that By Remark 1 , it is enough to prove that L n + k 0 has full range.Indeed, So, from Lemma 11 for C = 0 , the matrix of the linear operator

Proofs of theorems 1, 2 and 3
Proof of Theorem 1 .The sufficient condition is direct since the vector field F n has a rational first integral and the near-identity change of variables and re-parameterization of the time-variable transform its integral first into a non-analytic function, it which is quotient of formal series.
Let prove the necessary condition.From Proposition 5 , after a polynomial change of variables (if necessary) is transformed into (2) .
From now on, without loss of generality, we will consider the vector field (2) with d a rational number and d > 1 since otherwise, d < −1 , the change (x, y, t ) → (−x, y, −t ) for n even, or (x, y, t ) → (x, −y, −t ) for n odd, transforms the system (2) into Also, the invariant curves y − x n +1 and y + x n +1 are transformed into y + x n +1 and y − x n +1 , respectively.
We write the positive rational number with gcd (p, q ) = 1 .Thus, d = p+ q p−q and F n has the irreducible rational first integral I rat = f p g q , where f = y − x n +1 and g = y + x n +1 are the irreducible invariant curves of F n .
From [6, Theorem 19] , the vector field F has an unique analytic irreducible invariant curve C starting with f .Therefore, C p is the unique analytic invariant curve starting with f p .Moreover, the formal invariant curves starting with f p are C p u with u unit function.
Let I an algebraic and non-analytic first integral of F .By Proposition 4 , we know that there exists a first integral of the form In particular, its lowestdegree quasi-homogeneous term is also zero.So, By continuity, it extends to g n 0 .
Hence f m 0 g n 0 = I rat and I = C p u g q + ••• with C analytic invariant curve starting with f, g q + • • • formal invariant curve and u unit function.
We express the positive rational number d−1 n +1 = a b with gcd (a, b) = 1 .According the value of d, we distinguish two cases:

for all j natural number
Under these conditions, by Theorem 9 , the vector field (2) is orbitally equivalent to G = F n + j> 0 G n + j , where G n + j = μ n + j D 0 with μ n + j = α n + j x n + j , with j not multiple of n + 1 .
If F has an algebraic and non-analytic first integral then G also has an algebraic and non-analytic first integral ˆ I .In this case, f is an invariant curve of G , thus f p is the unique invariant curve starting with f p .Therefore, ˆ I = f p ḡ with ḡ = g q + j>q (n +1) ḡ j .
On the other hand, G ( By Proposition 4 , f p and ḡ have the same cofactor, i.e.
For proving the necessary condition enough to check that all μ n + j are zero.We imagine by reduction to the absurd one, that there exists a j 0 = min { j ∈ N , μ n + j = 0 } , i.e. μ n + j 0 = α n + j 0 x n + j 0 , with α n + j 0 = 0 and j 0 is not multiple of n + 1 .
Denoting m = j 0 + q (n + 1) , the equation ( 6) to degree j 0 + q (n + 1) Using the Euler Theorem for quasi-homogeneous polynomials, D 0 (p j ) = jp j with p j ∈ P t j , and re-sorting the terms, the equation ( 6) to order j 0 + q (n + 1) i.e. ḡ m satisfies the functional equation where (1)  n + m is the linear operator given in Lemma 13 .If ( a is odd) or ( a even and j 0 = b(2 n + 1) ), from Lemma 13 , the rightside of (7) belongs to a complementary subspace of the range of ˆ n + m therefore we arrive to contradiction, i.e. α n + j 0 = 0 .Otherwise, a even and j 0 = b(2 n + 1) , thus F is orbitally equivalent to G = F n + α n + j 0 x n + j 0 D 0 + • • • .The vector field G has the invariant curves (y − x n +1 ) p and (y + x n +1 ) q + • • • .We note that the curve (y − x n +1 ) p is the unique invariant curve starting with (y − x n +1 ) p and its cofactor is (n + 1) px n ((d − 1) + α n + j 0 x j 0 ) .The curve (y + x n +1 ) q + • • • could be not unique, i.e. could exist other invariant curve of the form (y + x n +1 ) q + • • • and its cofactor would be (n + 1) qx n ((d + 1) + α n + j 0 x j 0 ) + • • • .Therefore, if both factors were equal, then the quasi-homogeneous terms of degree n + j 0 would be equal, thus α n + j 0 = 0 .For proving the necessary condition enough to check that all G n + j are zero.We assume by reduction to the absurd one that there exists a j 0 = min { j ∈ N , G n + j = 0 } .We distinguish three cases: • If j 0 < k 0 , from Theorem 10 , the first term G n + j non-zero with j > 0 is G n + j 0 := α n + j 0 x n + j 0 (x∂ x + (n + 1) y∂ y ) , with α n + j 0 = 0 .
If k 0 is even, then a = 1 odd.So, by reasoning as before, by Lemma 13 , a complementary subspace to the range of the operator linear (1)  n + j 0 +(n +1) q is x j 0 + n g q , we arrive to contradiction.
• We suppose that j 0 = k 0 , the first term G n + j non-zero with the unique analytic invariant curve starting with f .From Proposition 7 , there exists a formal invariant curve C = f + c n +2 x n +2 + c n +3 x n +3 + • • • and its cofactor is x n .We now compute the first coefficient It is easy to check that c j+ n +1 = 0 , 1 That is, and sorting the coefficients, we get [2( By Proposition 4 , the vector field has an algebraic first integral if C p and ḡ = g q + j>q (n +1) ḡ j have the same cofactor, i.e.
The above equation (8) to degree m + n with m = k 0 + q (n + 1) is Using the Euler Theorem for quasi-homogeneous polynomials and re-sorting the terms, we have that ḡ m satisfies the functional equation If k 0 is even, then a = 1 odd.So, by Lemma 13 , a complementary subspace to the range of the operator linear (1)  n + k 0 +(n +1) q is x k 0 + n g q .Moreover, the right-hand of (9) belongs to Cor ( (1)  n + m ) .Indeed, following Lemma 12 , it is enough to prove that . Substituting, we get A = −(n + 1) 2 β n + k 0 .Thus, if β n + k 0 = 0 , we have that the right-hand of (9) belongs to Cor ( (1)  n + m ) and we arrive to contradiction.Therefore, β n + k 0 = 0 .
• if j 0 > k 0 , from Theorem 10 , an orbitally equivalent normal form of F is G = F n + j> 0 G n + j , where G n + j = α n + j x n + j (x∂ x + (n + 1) y∂ y ) , with j is not multiple of n + 1 .By reasoning as the case j 0 < k 0 , we prove that all the G n + j ≡ 0 .
Proof of Theorem 2 .We see the necessary condition.If the vector field F is analytically integrable then there exists , a polynomial change of variables, such that the lowest-degree quasihomogeneous term * F := F r is polynomially integrable.We distinguish three cases: for F r Hamiltonian vector field, we apply [1, Theorem 3.19] ; for F r non-Hamiltonian and irreducible, we apply [3, Theorem 1.2] ; and for F r non-Hamiltonian and reducible we apply [2,Theorem 1.4] .In all the cases, the analytic integrability leads to F orbitally equivalent to F r .
Otherwise, if F has an algebraic first integral but it is not analytic, we apply Theorem 1 .
The sufficient condition is trivial.Proof of Theorem 3 .By Proposition 5 , we can assume that, after a polynomial change of variables and a scaling, if necessary, F = F r + • • • where F r = P r+ t 1 ∂ x + Q r+ t 2 ∂ y is one of the following vector fields: For the case A) , the vector is a perturbation of the Hamiltonian vector field whose Hamiltonian function is
Fixed C ∈ R , we consider the linear operator ˆ n + j : P t j → P t n + j (10) p j ˆ n + j (p j ) = (F n + Cx n D 0 )(p j ) We have the following result.
Proof.We distinguish two cases: Assume i 0 = 0 , i.e. j = l 0 (n + 1) .In this case, P t j = g So, if there exists any d i 0 null, the range of the matrix is l 0 .
Therefore, the dimension of Cor ( ˆ n + j ) is one and we can choose Cor ( ˆ n + j ) = x j+ n since the elements e 0 , • • • , e l 0 −1 are all non-zero.
Otherwise, if all d i are non-zero, the operator has full range, that is Cor ( ˆ n + j ) is a trivial set.Now assume 0 < i 0 < n + 1 .In this case, P t j = x i 0 g i , i = 0 , • • • , l, do not belong to Range ( ˆ n + j ) .Thus, any one of them is a base of Range ( ˆ n + j ) .
We use the following result in the proof of Theorem 1 .

(l 0
) i , i = 0 , • • • , l 0 and P t n + j = x n g (l 0 ) i , i = 0 , • • • , l 0 .ifwe choose these basis, from Lemma 11 , the matrix of the linear operator ˆ n + j is the following square matrix and diagonal whose elements d i are all different +1) ×(l 0 +1)
the vector field F is analytically integrable if, and only if, it is orbitally equivalent to y∂ x + ax 2 n ∂ y .This vector field has the inverse integrating factor 2 ax 2 n +1 − (2 n + 1) y 2 .Thus, undoing the change of variables, it has a formal inverse integrating factor starting with 2 ax 2 n +1 − (2 n + 1) y 2 .For the case B) , we have t 1 xP r+ t 1 − t 2 yQ r+ t 2 = y (x n +1 − (n + 1) y ) .From [2, Theorem 1.5] , the vector field F is analytically integrable if, and only if, it has a formal inverse integrating factor starting with (n + 1) y (x n +1 − y ) .For the case C) the polynomialt 1 xP r+ t 1 − t 2 yQ r+ t 2 is (n + 1)(y 2 − x 2 n +2 ) .From [5,Theorem 11], these vector fields are orbitally equivalent to F r if, and only if, they have a formal inverse integrating factor starting with (n + 1)(y 2 − x 2 n +2 ) .So, from Theorem 2 , the result follows.Notice that, by Proposition 5 , after a polynomial change of variables, if necessary, we can always assume that if the leader quasihomogeneous term F r is rationally integrable then it satisfies that t 1 xP r+ t 1 − t 2 yQ r+ t 2 has only simple factors on C [ x, y ] .