A family of isochronous foci with Darboux first integral

Abstract

We consider the class of polynomial differential equations ? = ?x-y +Pn(x,y)+P2n-1(x,y), ? = x + ?y + Qn(x,y) + Q2n-1(x,y) with n = 2, where Pi and Qi are homogeneous polynomials of degree i. These systems have a focus at the origin if ?0, and have either a center or a focus if ? = 0. Inside this class we identify a new subclass of Darboux integrable systems having either a focus or a center at the origin. Under generic conditions such Darboux integrable systems can have at most two limit cycles, and when they exist are algebraic. For the case n = 2 and n = 3 we present new classes of Darboux integrable systems having a focus.