Orbital invariants of noncompact integrable systems with two degrees of freedom

We consider integrable Hamiltonian systems (IHS) with two degrees of freedom containing noncompact fibers under the condition of energy constancy. For such systems, analogues of the edge orbital invariant are studied, classifying IHSs on regular, topologically stable Liouville foliation sections with noncompact fibers (planes and cylinders) with respect to orbital equivalence. It turned out that generally for cylindrical fibers such invariant is the number of fibers with closed trajectories. If the fibers of two systems are homeomorphic to the plane, then the systems are smoothly conjugate on the corresponding regular parts of the Liouville foliation.