Quartic Rigid Systems in the Plane and in the Poincaré Sphere

We consider the planar family of rigid systems of the form x′=-y+xP(x,y),y′=x+yP(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'=-y+xP(x,y), y'=x+yP(x,y)$$\end{document}, where P is any polynomial with monomials of degree one and three. This is the simplest non-trivial family of rigid systems with no rotatory parameters. The family can be compactified to the Poincaré sphere such that the vector field along the equator is not identically null . We study the centers, singular points and limit cycles of that family on the plane and on the sphere.