FAMILIES OF TRANSVERSE POINCARÉ HOMOCLINIC ORBITS IN 2N-DIMENSIONAL HAMILTONIAN SYSTEMS CLOSE TO THE SYSTEM WITH A LOOP TO A SADDLE-CENTER

We prove the theorem: if an n-degrees-of-freedom Hamiltonian system has an equilibrium of the saddle-center type (there is a pair of simple eigenvalues ±iω; the rest of the spectrum consists of eigenvalues with nonzero real parts) with a homoclinic orbit to it then this system, and all those close to it, have transversal Poincaré homoclinic orbits to Lyapunov periodic orbits if some genericity conditions are satisfied. These conditions are pointed out explicitly. Thus a new criterion of nonintegrability has been obtained.