On the cubic perturbations of the symmetric 8-loop Hamiltonian

We study arbitrary cubic perturbations of the symmetric 8-loop Hamiltonian, which are linear with respect to the small parameter. It is shown that when the first 4 coefficients in the expansion of the displacement functions corresponding to both period annuli inside the loop vanish, the system becomes integrable with the following three strata in the center manifold: Hamiltonian, reversible in $y$ and reversible in $x$. In the latter case, the first integral is of Darboux type and we calculate it explicitly. Next we prove that the cyclicity of each of period annuli inside the loop is five and the total cyclicity of both is at most nine. For this, we use Abelian integrals method together with careful study the geometry of the separatrix solutions of related Riccati equations in connection to the second-order Melnikov functions.