  We prove that certain non-exact magnetic Hamiltonian systems on products of
closed hyperbolic surfaces and with a potential function of large oscillation
admit non-constant contractible periodic solutions of energy below the Ma\~n\'e
critical value. For that we develop a theory of holomorphic curves in
symplectizations of non-compact contact manifolds that arise as the covering
space of a virtually contact structure whose contact form is bounded with all
derivatives up to order three.
