Integrability, degenerate centers, and limit cycles for a class of polynomial differential systems

We consider the class of polynomial differential equations x˙ Pn(x,y)+Pn+1(x,y)+Pn+2(x,y), y˙=Qn(x,y)+Qn+1(x,y)+Qn+2(x,y), for n ≥ 1 and where Pi and Qi are homogeneous polynomials of degree i These systems have a linearly zero singular point at the origin if n > 2. Inside this class, we identify a new subclass of Darboux integrable systems, and some of them having a degenerate center, i.e., a center with linear part identically zero. Moreover, under additional conditions such Darboux integrable systems can have at most one limit cycle. We provide the explicit expression of this limit cycle.