The method of Puiseux series and invariant algebraic curves

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.