Isochronicity for a class of reversible systems

This paper deals with the relation between isochronicity and first integral for a class of reversible systems: x ˙ = − U ( x ) y , y ˙ = f ( x , y ) , which associates to the first integral of the form H ( x , y ) = F ( x ) y 2 + G ( x ) . Two necessary and sufficient conditions are given to characterize isochronicity for these systems. Moreover, we apply these results to show that there exists a class of polynomial reversible systems of degree n with isochronous center for any n.