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

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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.

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