Darboux integrability and Algebraic limit cycles for a class of polynomial differential Systems

Agraïments: The first and third authors are partially supported by NNSF of China grant No. 10671123 and by a CICYT grant No. 2005SGR 00550. The third author is also partially supported by NCET of China grant No. 050391.This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems ˙x = λx − y + Pn+1(x, y) + xF2n(x, y), ˙y = x + λy + Qn+1(x, y)+yF2n(x, y), where Pi(x, y), Qi(x, y) and Fi(x, y) are homogeneous polynomials of degree i. Inside this class we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 we present necessary and sufficient conditions for being Darboux integrable