Abstract

We consider planar polynomial differential systems. Suppose that $f(x,y)=0$ is an algebraic solution, where $f(x,y)$ is linear with respect to one of the variables $(x,y)$. We present a method that allows to employ this algebraic solution for obtaining information about the existence of other algebraic invariant curves or exponential factors of the system. In some cases, it is possible to prove easily with this method that a given polynomial system is not Darboux integrable. We apply this technique to prove that quadratic systems with an algebraic limit cycle given by a conic and the Yablonski family of quadratic systems with an algebraic limit cycle are not Darboux integrable.