The generalized Liénard polynomial differential systems x'=y,y'= -g(x) - f (x)y with deg g = deg f 1 are not Liouvillian integrable

Abstract

Agraïments: The second author was supported by Portuguese National Funds through FCT - Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD (PEst-OE/EEI/LA0009/2013)

We prove the nonexistence of Liouvillian first integrals for the generalized Li\'enard polynomial differential systems of the form x' = y, y'=-g(x)-f(x)y, where g(x) and f(x) are arbitrary polynomials such that g = f 1.