POLYNOMIAL AND RATIONAL FIRST INTEGRALS FOR PLANAR QUASI – HOMOGENEOUS POLYNOMIAL DIFFERENTIAL SYSTEMS

In this paper we find necessary and sufficient conditions in order that a planar quasi–homogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasi–homogeneous polynomial differential system can be transformed into a differential system of the form u̇ = uf(v), v̇ = g(v) with f(v) and g(v) polynomials, and vice versa.


Introduction
The characterization of polynomial or rational integrability of a differential system goes back to Poincaré, see [20,21,22] and has attracted the attention of many authors, see for instance [2,3,8,14,17,18,19,23,24] and references therein.For quasi-homogeneous polynomial differential systems if we control the polynomial first integrals we are controlling all analytical first integrals of the system, see [15,17].
We assume that there exists an analytic first integral H for an analytic differential system of the form ẋ = P (x, y), ẏ = Q(x, y).The analytic functions H, P and Q can be decomposed in sum of quasi-homogeneous polynomials of the same weight degree, i.e.H = H m + H m+1 + . .., P = P r + P r+1 + . . .and Q = Q r + Q r+1 + . ...Then, the quasi-homogeneous polynomial of the lowest weight degree H m must be a first integral of the quasi-homogeneous differential system ẋ = P r (x, y), ẏ = Q r (x, y), see [12,16].So the study of the integrability of the quasi-homogeneous polynomial differential systems is a good first step for studying the integrability of more general differential systems, see for instance [1,16].
Some links between Kowalevskaya exponents of quasi-homogeneous polynomial differential systems and the degree of their quasi-homogeneous polynomial first integrals are established in [6,9,17,24].constant on each trajectory of the system contained in U. We note that if H is of class at least C 1 in U, then H is a first integral if it is not locally constant and P (x, y) ∂H ∂x + Q(x, y) ∂H ∂y ≡ 0 in U. We call the integrability problem the problem of finding such a first integral and the functional class where it belongs.We say that the system has a polynomial first integral if there exists a first integral H(x, y) ∈ C[x, y].Analogously, we say that the system has a rational first integral if there exists a first integral H(x, y) ∈ C(x, y).
We say that a function V : W ⊆ C 2 → C, with W an open set, is an inverse integrating factor of system (1) if V is of class C 1 , not locally zero and satisfies the following linear partial differential equation where (x 0 , y 0 ) ∈ U is any point.An easy computation shows that the polynomial V (x, y) = s 2 yP (x, y) − s 1 xQ(x, y) is an inverse integrating factor for the quasi-homogeneous system (1) of weight (s 1 , s 2 , d).
A proof of this well-known fact can be found in Proposition 15 of [8].This result generalizes the particular case for homogeneous systems that can be found in Lemma 3 of [4].Therefore the quasi-homogeneous systems (and in particular the homogeneous ones) have always a polynomial inverse integrating factor.In [5] necessary conditions were given in order to have a polynomial inverse integrating factor for general polynomial differential systems.To see the relation between the functional classes of the inverse integrating factors and their associated first integrals see Theorem 3 of [11].
Our main results are stated in Lemma 2, Theorem 9 and Proposition 11.Theorem 9 characterizes when a quasi-homogeneous polynomial differential system (1) has either a polynomial, or a rational first integral.In Lemma 2 and Proposition 11 we show that any planar quasi-homogeneous polynomial differential system can be transformed into a differential system of the form u = uf (v), v = g(v) with f (v) and g(v) polynomials, and vice versa.The rest of the paper is organized as follows.In section 2 we present some lemmas (including Lemma 2) which will allow us to prove Theorem 9 and Proposition 11 in section 3.

Preliminary results
Lemma 1.Given a quasi-homogeneous system (1) of weight (s 1 , s 2 , d), we can suppose without restriction that s 1 and s 2 are coprime.
Proof.Let r be the maximum common divisor of s 1 and s 2 .Then s 1 = rs * 1 and s 2 = rs * 2 with s * 1 and s * 2 coprime.Let x i p y j p be a monomial with nonzero coefficient of P (x, y).We know its existence because P and Q are coprime.We also know that then we consider a monomial x iq y jq with nonzero coefficient of Q(x, y).Taking into account that From here we deduce that d − 1 is divisible by r except if (i q , j q ) = (0, 1).Therefore the unique case to study is P (x, y) = x and Q(x, y) = y, i.e. ẋ = x and ẏ = y, which is a homogenous system of degree 1 with (s 1 , s 2 , d) = (1, 1, 1).We have excluded linear systems from consideration.In any other case we have that d − 1 is divisible by r and we can write d − 1 = r * (d * − 1).
In short, we claim that system (1) is (s * 1 , s * 2 , d * ) quasi-homogeneous with s * 1 and s * 2 coprime.Indeed, we have that the monomial x i p y j p of P (x, y) must verify that s 1 (i p − 1) + s 2 j p = d − 1, but substituting In a similar way for any monomial x iq y jq of Q(x, y) we obtain Hence we obtain a quasi-homogeneous system of weight (s * 1 , s * 2 , d * ).Lemma 2. The change of variables and the rescaling of time given by u , with m ∈ N ∪ {0}, transforms a quasi-homogeneous system (1) of weight (s 1 , s 2 , d) into a polynomial system of the form Moreover we can choose m in such a way that the polynomials f (v) and g(v) are coprime.
Notice that if we take α = u ) and by the quasi-homogeneity we have In a similar way and also by quasi-homogeneity we have Hence system (6) becomes (7) u = s 2 u ] .
Now we divide the system by u and we have ] .
We know that the monomials x ip y jp of P (x, y) and the monomials x iq y jq of Q(x, y) satisfy Hence u has the monomials in v of the form v m+1+j p −s 1 s 1 , and v has the monomials in v of the form or v m+j q s 1 .The identities (9) modulo s 1 are of the form (10) If s 1 and s 2 are coprime, i.e. (s 1 , s 2 ) = 1, then there exists s −1 2 ∈ Z such that s −1 2 s 2 = 1 (mod s 1 ).Since s 1 and s 2 are coprime, by the Bézout identity there exist two integers x and y such that s 1 x + s 2 y = 1.Hence, modulo s 1 we have s 2 y = 1 and we take s −1 2 = y.Therefore the identities (10) can be simplified multiplying by s −1 2 and we obtain and consequently j p ≡ j q − 1 (mod s 1 ) for all j p and j q .We define m ∈ N∪{0} the smaller value such that m+1+j p ≡ m+j q ≡ 0 (mod s 1 ), and with this choice all the monomials that appear in system (8) have nonnegative integer exponents and consequently system (8) is polynomial and of the form u = uf (v) and v = g(v).We need to see that the obtained system (8) is coprime.Note that for the choice of m, being the smallest one, it cannot happen that v|f (v) and v|g(v), because in this case we could choose m even smaller.
Let ξ ∈ C such that P (1, ξ 1 s 1 ) = 0, i.e. ξ is a root of f (v).We are going to see that this root ξ is not a root of g(v).If g(ξ) = 0 then we have that −s 2 ξ s 1 ) = 0. Now we consider the algebraic curve y s 1 − ξx s 2 = 0, and we parameterize it as follows x = α s 1 , y = α s 2 ξ 1/s 1 , with α ∈ C. By quasi-homogeneity and taking into account that P (1, ξ Hence, P (x, y) and Q(x, y) vanish at the same place.Therefore by the Bézout theorem, P and Q would have a common factor, in contradiction with the initial hypothesis that P and Q were coprime.We observe that this argument does not hold in the case that f (v) is a constant different from zero and g(v) ≡ 0. In this case, system (1) is linear, see Lemma 3, and linear systems have been excluded from consideration.
One immediate consequence of Lemma 2 is that if system (5) has H(u, v) as a first integral, then the quasi-homogeneous system (1) of weight (s 1 , s 2 , d) has a first integral of the form H(x s 2 , y s 1 /x s 2 ).On the other hand, if H(x, y) is a first integral of the quasi-homogeneous system (1) of weight (s 1 , s 2 , d) then H(u 1/s 2 , (u, v) 1/s 1 ) is a first integral of system (5).Lemma 3. Consider system (1) with P and Q coprime.The particular case when f (v) is constant different from zero and g(v) ≡ 0 corresponds to the system ẋ = s 1 x and ẏ = s 2 y.
Proof.The condition g(v) ≡ 0 implies that the inverse integrating factor V (x, y) ≡ 0, i.e. s 2 yP (x, y) − s 1 xQ(x, y) ≡ 0. In this case P (x, y) is divisible by x due to P and Q are coprime.Hence, we can find a polynomial M (x, y) such that P (x, y) = s 1 xM (x, y) and consequently Q(x, y) = s 2 yM (x, y).However as P and Q are coprime we have that M (x, y) must be a constant different from zero, and with a timerescaling we obtain the system ẋ = s 1 x and ẏ = s 2 y.This linear system has the rational first integral H = x s 2 y −s 1 and has not any polynomial first integral.Now we are going to study the polynomial and rational integrability of system (5).
Lemma 4. Consider a polynomial system of the form (5), where f (v) and g(v) are coprime.
then it has a first integral of the form H(u, v) = h(v)u s where h(v) is a non constant rational (resp.polynomial) function and (ii) Assume that system (5) has a rational or polynomial first integral of the form where A and B are coprime polynomials that can be written in powers of u into the form with n, m ∈ N ∪ {0} and a n (v) b m (v) ̸ = 0.The polynomial case corresponds to m = 0 and b 0 (v) = 1.
In the particular case that n = m and if there exists k ∈ C such that a n (v) = kb m (v) we take as a first integral H(u, v) − k instead of H(u, v) and we have In the polynomial case, if n = m = 0 then we will have that H(u, v) = a 0 (v) which implies v = 0, and this gives a contradiction because f (v) and g(v) are coprime.We can assume that n ≥ m, because if H is a first integral of system (5) then 1/H is also a first integral, and for the case n = m, we can assume that a n (v)/b n (v) is not constant as we have shown in the previous paragraph.If H is a first integral of system (5) then it must satisfy (12) uf Substituting H = A/B in (12) and multiplying by B 2 we have We now consider the highest degree coefficient in this expression, which corresponds to u n+m and we obtain that This identity says that u n−m a n (v)/b m (v) is a first integral of system (5).Therefore we have a first integral of the form is rational when H is rational, and is polynomial when H is polynomial.Note that h(v) cannot be a constant and s must be different from zero because both cases are in contradiction with the hypothesis that f (v) and g(v) are coprime.Hence we have proved statement (ii).
(iii) We assume that system (5) has a rational first integral and we take the first integral of the form and we have the following differential equation Now we assume that deg f ≥ deg g and we consider the Euclidean division of −sf (v) by g(v), so we have where ψ(v) cannot be zero taking into account that f and g are coprime and deg ψ < deg g.Hence equation ( 13) takes the form .
Integrating this differential equation we have where C is a constant of integration and q ′ (v) = q(v).Therefore the first factor of (15) cannot cancel with the second factor of ( 15) and this gives a contradiction with the fact that h(v) is a rational function.
Hence, we conclude that deg f < deg g.
(iv) From the proof of statement (iii) we have obtained the differential equation (13).We assume now that V (u, v) is not square-free.Using an affine transformation of the form v → v + α with α ∈ C if it is necessary, we can assume that v is a multiple factor of V (u, v) with multiplicity µ > 1.Then we have that V (u, v) = ug(u) = uv µ r(v) with r(0) ̸ = 0. We know that f (0) ̸ = 0 because f and g are coprime.Now we develop the right-hand side of (13) in simple fractions of v, that is, where α 0 (v) and α 1 (v) are polynomials with deg α 1 (v) < deg r(v) and c i ∈ C, for i = 1, 2, . . ., µ. Equating both expressions, we get that c µ = −s f (0)/r(0) ̸ = 0. Moreover as a consequence of statement (iii) we know that α 0 (v) ≡ 0. Therefore equation ( 13) becomes with c µ ̸ = 0. Now if we integrate this expression we get where C is a constant of integration.The first exponential factor cannot be simplified with any part of the second exponential factor.Thus, we get a contradiction with the fact that h(v) is a rational function.
Therefore we conclude that V (u, v) and g(v) are square-free.This proves statement (iv).
(vi) We must prove that To see that this last expression is identically zero is equivalent to see that . Recalling the expression of φ(v) we have Taking common denominator and recalling that g

Now substituting the values of γ
Since deg f < deg g, the expression in the sum is the Lagrange polynomial which interpolates the k points (α i , f (α i )), for i = 1, 2, . . ., k, see for more details [13].Therefore, this polynomial is f (v) and we conclude that expression ( 16) is identically satisfied.
(vii) We assume that system (5) has a rational first integral, then by statement (ii) it has one of the form h(v)u s where h(v) satisfies identity (13).Using statements (iii), (iv) and (vi) we have that ( 17) Hence, integrating identity (13) we obtain ( 18) Then, as h(v) must be a rational function and s ∈ N we have that . ., k the first integral described in statement (vi) elevated to certain natural power leads to a rational first integral of system ( 5).
(viii) If system (5) has a polynomial first integral, then reasoning as in the previous statement we arrive to the same expression for h(v) given by ( 18) which is polynomial if and only if . ., k the multiplicative inverse of the first integral described in statement (vi) gives a polynomial first integral of system (5).This completes the proof of the lemma.
In order to relate the first integrals of the quasi-homogeneous system (1) of weight (s 1 , s 2 , d) with the first integrals of system (5), we must see how the change of variables described in Lemma 2 affects to the rationality/polinomiality of a first integral of system (1).First we need an auxiliary lemma about quasi-homogeneous polynomials.
Lemma 5.The following statements hold.
(i) Let A ℓ (x, y) be a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree ℓ, then ∂A ℓ /∂x is a quasihomogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree ℓ − s 1 , and ∂A ℓ /∂y is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree ℓ − s 2 .(ii) Let A ℓ (x, y) and B m (x, y) be two quasi-homogeneous polynomials of weight exponents (s 1 , s 2 ) and of weight degrees ℓ and m, respectively, then their product is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree ℓ + m and their quotient is a quasi-homogeneous function of weight exponents (s 1 , s 2 ) and of weight degree ℓ − m. (iii) Let A(x, y) be a polynomial.Then A(x, y) can be written as an ordered finite sum of quasi-homogeneous polynomials of weight exponents (s 1 , s 2 ) and of weight positive degrees, i.e.A(x, y) = A 0 (x, y) + A 1 (x, y) + • • • + A ℓ (x, y) where A i (x, y) is a quasihomogeneous polynomials of weight exponents (s 1 , s 2 ) and of weight degree i.
Proof.Although the proof is well-known, we give it here for sake of completeness.
(i) First we recall the definition of quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) given in the introduction.We have that A ℓ (x, y) is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree ℓ if Hence, if x i y j is a monomial with a nonzero coefficient of A ℓ (x, y) we have α s 1 i x i α s 2 j y j = α ℓ x i y j from here we obtain s 1 i + s 2 j = ℓ for all i, j.The polynomial ∂A ℓ /∂x has the monomial ix i−1 y j (except for the case i = 0) and using the same reasoning we will obtain the relation s 1 (i − 1) + s 2 j.In fact taking into account that s 1 i + s 2 j = ℓ we get s 1 (i − 1) + s 2 j = ℓ − s 1 which implies that the polynomial ∂A ℓ /∂x is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree ℓ − s 1 .The proof for ∂A ℓ /∂y is analogous.
(ii) We consider the product polynomial and we evaluate at (α s 1 x, α s 2 y) and we have The same reasoning is valid for the quotient.
(iii) Let x i y j be a monomial with a nonzero coefficient of A(x, y).It is clear that belongs to a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree s 1 i + s 2 j.Since i, j ≥ 0 we have that this degree is not negative.As the number of monomials of A(x, y) is finite we obtain a finite number of quasi-homogeneous polynomials of weight exponents (s 1 , s 2 ) and the weight degrees are ordered.Lemma 6.If system (1) has a rational (resp.polynomial) first integral then system (1) has has a rational (resp.polynomial) first integral which is a quasi-homogeneous function of weight exponents (s 1 , s 2 ) and of weight degree m ≥ 0.
Proof.The polynomial case could be deduced from the Proposition 1 in [17].
We consider H(x, y) = A(x, y)/B(x, y) the rational or polynomial first integral of system (1) and, using Lemma 5 statement (iii) we write A(x, y) and B(x, y) as a sum of quasi-homogeneous polynomials of weight exponents (s 1 , s 2 ) where a, b ∈ N ∪ {0} and A a (x, y)B b (x, y) ̸ = 0. Note that the polynomial case is b = 0 and B 0 (x, y) = 1.In the particular case a = b and if there exists k ∈ C such that A a (x, y) = kB a (x, y) we take H(x, y) − k instead of H(x, y) in order to have a first integral where the degree of the numerator is less than the degree of the denominator.We remark that if H is a first integral then 1/H is also a first integral.Therefore we can assume that a ≥ b and in the particular case that a = b we can assume that the quotient A a (x, y)/B a (x, y) is not constant.If H = A/B is first integral of system (1) it must satisfy Substituting H = A/B and multiplying by B 2 we have ( 19) Taking into account Lemma 5 we have that P is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree s 1 − 1 + d and Q is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree s 2 −1+d, and the polynomial P ∂A i /∂x+Q∂A i /∂y is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree d−1+i for i = 0, 1, 2, . . ., a and the same happens for the polynomial P ∂B i /∂x+Q∂B i /∂y for i = 0, 1, 2, . . ., b. Taking the terms of highest degree in (19), which correspond to the quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree d−1+a+b we obtain This equality says that A a (x, y)/B b (x, y) is a first integral of system (1).The quotient of two quasi-homogeneous polynomials of weight exponents (s 1 , s 2 ) is a quasi-homogeneous function of weight exponents (s 1 , s 2 ) and of weight degree the difference between the weight degrees of the two polynomials (see statement (ii) of Lemma 5).Hence, the rational function A a (x, y)/B b (x, y) is a quasi-homogeneous function of weight exponents (s 1 , s 2 ) and of weight degree a − b ≥ 0.
The following lemma shows the relationship between first integrals of system (1) and the ones of system (5).Lemma 7. The following statements hold.
(i) System (1) has a rational first integral if and only if system (5) has a rational first integral.(ii) If system (1) has a polynomial first integral, then system (5) has a polynomial first integral.(iii) If system (5) has a polynomial first integral, then system (1) has a first integral of the form x ℓ p(x, y) where p(x, y) is a (s 1 , s 2 ) quasi-homogeneous polynomial and ℓ ∈ Z.
Proof.In all this proof we assume that s 1 and s 2 are coprime using Lemma 1.
(i) We assume that system (1) has a rational first integral, then using Lemma 6, we know that it has a first integral which is a quasihomogeneous function of weight exponents (s 1 , s 2 ) and of the form A a (x, y)/B b (x, y) where A a (x, y) is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree a and B b (x, y) is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree b.We consider the first integral where L ∈ N is such that the degree of the numerator and the denominator is a multiple of s 1 s 2 .Hence, we can assume that we have a rational first integral of system (1) of the form A(x, y)/B(x, y) where A(x, y) and B(x, y) are quasi-homogeneous polynomials of weight exponents (s 1 , s 2 ) and of weight degree a and b respectively where a = s 1 s 2 ã and b = s 1 s 2 b with ã and b ∈ N. Let x i y j be a monomial with a nonzero coefficient of A(x, y) (or B(x, y)).By quasi-homogeneity we have As s 1 and s 2 are coprime, we deduce that j is a multiple of s 1 , that is, j = s 1 j with j ∈ N ∪ {0}.The change of variables described in Lemma 2 tells us that if H = A(x, y)/B(x, y) is a first integral of system (1), then H(u ) is a first integral of system (5).Moreover we recall that H(x, y) is a quasi-homogeneous function of weight exponents (s 1 , s 2 ) and of weight degree a − b = s 1 s 2 (ã − b), and we shall denote by m = ã − b.We take α = u 1 s 1 s 2 and by quasi-homogeneity we have where m ∈ Z.As all the monomials x i y j that appear in H(1, v 1 s 1 ) have j multiple of s 1 , we have that H(1, v 1 s 1 ) is a rational function of v. Therefore we have a first integral of system (5) of the form u mh(v) with h(v) a rational function and m ∈ Z.
Conversely, we assume that system (5) has a rational first integral H(u, v).By the change of variables described in Lemma 2 we have that H(x s 2 , y s 1 /x s 2 ) is a first integral of system (1) which clearly is a rational function.
(ii) We assume that system (1) has a polynomial first integral.By similar arguments used in statement (i) we can assume that system (1) has a polynomial first integral H(x, y) which is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree m = s 1 s 2 m with m ∈ N. Analogously as in the previous section, we can see that all the monomials x i y j that appear in H(x, y) have j multiple of s 1 , i.e. j = js 1 with j ∈ N ∪ {0}.Moreover, we have that H(u is a first integral of system (5) and, by quasi-homogeneity, as before, we can see that As H(x, y) is a polynomial and all the monomials x i y j satisfy that j is a multiple of s 1 , we have that h(v) = H(1, v 1 s 1 ) is a polynomial in v. Therefore, we have that u mh(v) is a polynomial first integral of system (5).
(iii) If system (5) has a polynomial first integral, using statement (ii) of Lemma 4, we know that it has a first integral of the form u s h(v) with s ∈ N and h(v) a polynomial.Therefore by the change of variables described in Lemma 2 we have that x s 2 s h(y s 1 /x s 2 ) is a first integral of system (1).As h(v) is a polynomial we can write it into the form We denote by ) , and we have that x s 2 (s−k) r(x, y) is a rational first integral of system (1).We note that r(x, y) is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) due to ) , which implies r(α s 1 x, α s 2 y) = α s 1 s 2 k r(x, y).
The following example shows that there are systems (5) having polynomial first integrals and such that their corresponding systems (1) have no polynomial first integral.
Its associated system (5) has a polynomial first integral, but system (21) has a rational first integral and has no polynomial first integrals.
Proof.System ( 21) is a quasi-homogeneous system of weight exponents (1, 1) and of weight degree 3. Applying the change of variables described in Lemma 2 given in this case by x = u and y = uv we obtain the system Comparing with system (5) we have that f 22) has the polynomial first integral H = u 2 v(v − 1)(v + 1) and undoing the change we obtain that system (21) has the rational first integral H(x, y) = y(y − x)(y + x) x .
Now we prove that system (21) has no polynomial first integrals.We assume that it has a polynomial first integral and by Lemma 6 it must have a homogeneous polynomial first integral H m (x, y) of degree m ≥ 0. Moreover as H m (x, y) is a first integral we have m > 0. Using the change of variables of Lemma 2 and applying the results of Lemma 7, system (21) would have a first integral of the form u m h(v) where h(v) will be a polynomial.We know that h(v), as we have seen in the proof of statement (iii) of Lemma 4, must satisfy the differential equation (13) and in this case we have ) ,

Solving this equation we obtain
where C is a constant of integration that must be different from zero that we take equal to 1 without loss of generality.Undoing the change x = u and y = uv, we obtain that H m (x, y) = x −m/2 y m/2 (y 2 − x 2 ) m/2 .Since m > 0 we obtain that H m (x, y) cannot be a polynomial and this gives a contradiction with the existence of a polynomial first integral for system (21).

The main results
The following theorem is the main result of this paper and characterizes when system (1) has a polynomial or a rational first integral.As usual Q denotes the set of rational numbers, and Q + (resp.Q − ) the set of positive (resp.negative) rational numbers.
Theorem 9. Consider system (1) which can be transformed by the change defined in Lemma 2 in system (5).Using the same notation than in previous lemmas, the following statements hold.
(a) System (1) has a rational first integral if and only if g(v) is square-free, deg f < deg g and Proof.(a) By statement (i) of Lemma 7 system (1) has a rational first integral if and only if, system (5) has a rational first integral.By statement (vii) of Lemma 4 system (5) has a rational first integral if and only if, g(v) is square-free, deg f < deg g and γ i ∈ Q for i = 1, 2, . . ., k.
(b) We assume that system (1) has a polynomial first integral.By statement (ii) of Lemma 7 we have that system (5) has a polynomial first integral.By statement (viii) of Lemma 4 we have that g(v) is square-free, deg f < deg g and γ i ∈ Q − for i = 1, 2, . . ., k.We need to see that 1 + We consider a polynomial first integral of system (1).By Lemma 6, we can suppose that this first integral is a quasi-homogeneous polynomial of weight exponents (s 1 , s 2 ) and of weight degree m ≥ 0 and as it is a nonconstant polynomial we have that m > 0. We denote this first integral by H m (x, y).As we have seen in the proof of statement (ii) of Lemma 7, we have a polynomial first integral of system (5) of the form u mh(v) with m > 0 and h(v) a polynomial.Moreover, as we have seen in the proof of statement (iii) of Lemma 4, the polynomial h(v) has the form where C is a constant of integration that we can take without loss of generality equal to 1.In fact, due to the change of variables defined in that jp > j p for any other monomial because s 1 i p > 0. Consequently the polynomial Moreover we observe that if ã is the coefficient of the monomial y j p of P (x, y) then we have f (v) = s 2 ãv k−1 + • • • , where the dots mean terms in v of lower degree.Let jq be the largest natural such that x ĩq y jq is a monomial with nonzero coefficient of Q(x, y).We denote by b the coefficient of this monomial.By the quasi-homogeneity of Q(x, y) we have and equating coefficients we get In Lemma 2 we have shown that any quasi-homogeneous polynomial differential system after a convenient change of variables can be transformed into a polynomial differential system of the form u = uf (v), , y) in W. The knowledge of an inverse integrating factor defined in W allows the computation of a first integral in U = W \ {V = 0} doing the line integralH(x, y) = ∫ (x,y) (x 0 ,y 0 ) P (x, y)dy − Q(x, y)dx V (x, y) ,