Reversal Symmetries for Planar Vector Fields

In this paper, we formulate a comprehensive study of relevant properties of reversible vector fields. As a consequence, we prove that the reversibility of the first non-zero quasi-homogeneous term, respect to some types, of a vector field is a necessary condition for the reversibility of the vector field. We also provide a straightforward characterization of the reversibility for quasi-homogeneous vector fields. Finally, as an application of our previous results, we analyze some special polynomial and nilpotent systems, including examples which are centers and non-reversible.