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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 Liénard 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 Liénard 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.