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

We consider the class of polynomial differential equations ẋ = Pn(x, y)+Pn+1(x, y) +Pn+2(x, y), ẏ = 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 1 limit cycle. We provide the explicit expression of this limit cycle.