On Lie's symmetries for planar polynomial differential systems

In this paper we show that any polynomial planar vector field which is eith er polynomial or rational integrable possesses a polynomial or rational inf initesimal generator of a Lie symmetry, respectively. Moreover, if all the critical points of the vector field are strong and there exists a polynomia l inverse integrating factor that vanishes at all the critical points we sh ow that, independently of the class of their first integral, there exists a polynomial infinitesimal generator of a Lie symmetry.