Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle

It is known that perturbations from a Hamiltonian 2-saddle cycle$\Gamma $can produce limit cycles that are not covered by theAbelian integral, even when the Abelian integral is generic. Theselimit cycles are called alien limit cycles. In this paper,extending the results of [6] and [2], we investigatethe number of alien limit cycles in generic multi-parameter rigidunfoldings of the Hamiltonian 2-saddle cycle, keeping oneconnection unbroken at the bifurcation.