Algebraic invariant curves for the Lienard equation

Odani has shown that if deg g less than or equal to deg f then after deleting some trivial cases the polynomial system (x) over dot = y, (y) over dot = -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.