Analytic integrability of two lopsided systems

In this paper, we present two classes of lopsided systems and discuss their analytic integrability. The analytic integrable conditions are obtained by using the method of inverse integrating factor and theory of rotated vector field. For the first class of systems, we show that there are n + 4 small-amplitude limit cycles enclosing the origin of the systems for n ? 2, and 10 limit cyclesfor n = 1. For the second class of systems, we prove that there exist n+4 small-amplitude limit cycles around the origin of the systems for n ? 2, and 9 limit cycles for n = 1.