Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider two-degree-of-freedom Hamiltonian systems with a saddle-center loop, namely an orbit homoclinic to a saddle-center equilibrium (related to pairs of pure real, (plus-minus)(nu), and pure imaginary, (plus-minus)(omega)i, eigenvalues). We study the topology of the sets of orbits that have the saddle-center loop as their (alpha) and (omega) limit set. A saddle-center loop, as a periodic orbit, is a closed loop in phase space and the above sets are analogous to the unstable and stable manifolds, respectively, of a hyperbolic periodic orbit.