Invariant surfaces and Darboux integrability for non-autonomous dynamical systems in the plane
A novel method of finding and classifying irreducible invariant surfaces of non-autonomous polynomial dynamical systems in the plane is presented. The general structure of irreducible invariant surfaces and their cofactors is found. The complete set of irreducible invariant surfaces for the classical forced Duffing-van der Pol oscillator is obtained. It is proved that the forced Duffing-van der Pol oscillator possesses only one independent generalized Darboux first integral provided that a constraint on the parameters is valid. In other cases generalized Darboux first integrals do not exist. Consequently, the forced Duffing-van der Pol oscillator is not integrable with two independent generalized Darboux first integrals.