The solution of the Poincaré problem on the rational first integral for the Liénard polynomial differential equations

Abstract

In this work we classify the polynomial Liénard differential equations ẍ+f(x)ẋ+x=0, having a rational first integral. Such classification was asked by Poincaré in 1891 for any general polynomial differential systems in the plane R2. As far as we know it is the first time that the complete classification is given for a relevant class of polynomial differential equations of arbitrary degree.