It is known that perturbations from a Hamiltonian 2-saddle cycle
$\Gamma $can produce limit cycles that are not covered by the
Abelian integral, even when the Abelian integral is generic. These
limit cycles are called alien limit cycles. In this paper,
extending the results of  and , we investigate
the number of alien limit cycles in generic multi-parameter rigid
unfoldings of the Hamiltonian 2-saddle cycle, keeping one
connection unbroken at the bifurcation.