Invariant algebraic curves for Liénard dynamical systems revisited

A novel algebraic method for finding invariant algebraic curves for a polynomial vector field in C2 is introduced. The structure of irreducible invariant algebraic curves for Liénard dynamical systems xt=y, yt=−f(x)y−g(x) with degg=degf+1 is obtained. It is shown that there exist Liénard systems that possess more complicated invariant algebraic curves than it was supposed before. As an example, all irreducible invariant algebraic curves for the Liénard differential system with degf=2 and degg=3 are obtained. All these results seem to be new.