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

We consider the class of polynomial differential equations x = lambdax - y + P-n(x, y), y = x + lambday + Q(n)(x, y), where P-n and Q(n) are homogeneous polynomials of degree n. These systems have a focus at the origin if lambda not equal 0, and have either a center or a focus lambda = 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. (C) 2003 Elsevier Inc. All rights reserved.