  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.
