Centers of reversible cubic perturbations of the symmetric 8-loop Hamiltonian

We show that the center set of reversible cubic systems, close to the symmetric Hamiltonian system $x^{\mathrm{\prime }}=y,y^{\mathrm{\prime }}=x-x^3$ has two irreducible components of co-dimension two in the parameter space. One of them corresponds to the Hamiltonian stratum, the other to systems which are polynomial pull back of an appropriate linear system