Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems

Agraïments: The first author is partially supported by a FAPESP-BRAZIL grant 2014/26149-3. The third author is partially supported by a FAPESP-BRAZIL grant 2012/18780-0 and a CNPq grant 449655/2014-8. The three authors are also supported by a CAPES CSF-PVE grant 88881.030454/ 2013-01 from the program CSF-PVE.In this paper we completely characterize trivial isochronous centers of degrees 5 and 7. Precisely, we provide formulas, up to linear change of coordinates, for the Hamiltonian H of the isochronous centers such that H =(f_1^2 f_2^2)/2 has degrees 6 and 8, and f = (f_1, f_2): R^2 R^2 is a polynomial map with D f = 1 and f(0,0) = (0,0)