  Applying techniques originally developed for systems lacking a variational
structure, we establish conditions for the existence of solutions in systems
that possess this property but their energy functional is unbounded both lower
and below. We show that, in general, our conditions differ from those in the
classical mountain pass approach by Ambroseti-Rabinovitz when dealing with
systems of this type. Our theory is put into practice in the context of a
coupled system of Stokes equations with reaction terms, where we establish
sufficient conditions for the existence of a solution. The systems under study
are intermediary between gradient-type systems and Hamiltonian systems
