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We study the existence of real-analytic first integrals and real-analyticintegrability for perturbations of integrable systems in the sense ofBogoyavlenskij including non-Hamiltonian ones. We especially assume that thereexists a family of periodic orbits on a regular level set of the firstintegrals having a connected and compact component and give sufficientconditions for nonexistence of the same number of real-analytic first integralsin the perturbed systems as the unperturbed ones and for their real-analyticnonintegrability near the level set such that the first integrals andcommutative vector fields depend analytically on the small parameter. Wecompare our results with classical results of Poincar\'e and Kozlov for systemswritten in action and angle coordinates and discuss their relationships withthe subharmonic and homoclinic Melnikov methods for periodic perturbations ofsingle-degree-of-freeedom Hamiltonian systems. We illustrate our theory forthree examples containing the periodically forced Duffing oscillator.

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