Algebraic invariant curves for the Liénard equation

Odani has shown that if $\deg g\leq \deg f$ then after deleting some trivial cases the polynomial system $\dot {x}=y,\,\,\dot {y}=-f(x)y-g(x)$ does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of $\deg f$ and $\deg g$. We give also a simple presentation of Yablonsky's example of a quartic limit cycle in a quadratic system.