Obstructions to Integrability of Nearly Integrable Dynamical Systems Near Regular Level Sets

We study the existence of real-analytic first integrals and integrability for perturbations integrable systems in sense Bogoyavlenskij, including non-Hamiltonian ones. In particular, we assume that there exists a family periodic orbits on regular level set having connected compact component give sufficient conditions nonexistence same number perturbed as unperturbed ones their nonintegrability near such commutative vector fields depend analytically small parameter. compare our results with classical Poincaré Kozlov written action angle coordinates discuss relationships subharmonic homoclinic Melnikov methods single-degree-of-freedom Hamiltonian systems. latter discussion reveals can be real-analytically nonintgrable even if no transverse orbit to orbit. illustrate theory three examples containing periodically forced Duffing oscillator.