Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations x˙=λx-y+Pn(x,y),y˙=x+λy+Qn(x,y), where Pn and Qn are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.