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

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.