Relationships between Darboux Integrability and Limit Cycles for a Class of Able Equations.

We consider the class of polynomial differential equation x=pn(x,y)+ p(n+m) (x,y) + p(n+m) (x,y) + p(n+2m) (x,y), y= qn(x,y) + q(n+m) + q(n+2m) (x,y). for m,n>=1 where p1 and q1 are homogeneous polynomials of degree i. inside this class of polynomial differential equation we consider a subclass of darboux integrable systems. moreover, under additional conditions we proved such darboux integrable systems can have at most 1 limit cycle.