When Jacobian matrices at two saddle-centers share the same pair of purely imaginary eigenvalues, transverse heteroclinic intersections on a common energy surface occur exactly under the paper's cited sufficient conditions for nonintegrability; otherwise intersections are independent of those条件.
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Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers
When Jacobian matrices at two saddle-centers share the same pair of purely imaginary eigenvalues, transverse heteroclinic intersections on a common energy surface occur exactly under the paper's cited sufficient conditions for nonintegrability; otherwise intersections are independent of those条件.