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. © 2003 Elsevier Inc. All rights reserved.