  We study the existence of real-analytic first integrals and real-analytic
integrability for perturbations of integrable systems in the sense of
Bogoyavlenskij including non-Hamiltonian ones. We especially assume that there
exists a family of periodic orbits on a regular level set of the first
integrals having a connected and compact component and give sufficient
conditions for nonexistence of the same number of real-analytic first integrals
in the perturbed systems as the unperturbed ones and for their real-analytic
nonintegrability near the level set such that the first integrals and
commutative vector fields depend analytically on the small parameter. We
compare our results with classical results of Poincar\'e and Kozlov for systems
written in action and angle coordinates and discuss their relationships with
the subharmonic and homoclinic Melnikov methods for periodic perturbations of
single-degree-of-freeedom Hamiltonian systems. We illustrate our theory for
three examples containing the periodically forced Duffing oscillator.
