An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Li\'enard dynamical systems $x_t=y$, $y_t=-f(x)y-g(x)$ with $\text{ deg}\, f<\text{ deg}\,g<2\,\text{ deg}\,f+1$. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Li\'enard dynamical systems with $\text{ deg}\, f<\text{ deg}\, g<2\,\text{ deg}\, f+1$ can have at most two distinct irreducible invariant algebraic curves simultaneously and consequently are not integrable with a rational first integral.
Categories: math.DS nlin.SI