Resumen de Liouvillian Integrability Versus Darboux Polynomials

Jaume Llibre , Claudia Valls , Xiang Zhang

• In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Liénard polynomial differential system x˙ = y, y˙ = − f (x)y −g(x) with deg f > deg g is not Liouvillian integrable.