We study the spectrum of a periodic self-adjoint operator on the
axis perturbed by a small localized nonself-adjoint operator. It
is shown that the continuous spectrum is independent of the perturbation,
the residual spectrum is empty, and the point
spectrum has no finite accumulation points. We study the existence of the
embedded eigenvalues. We establish the
necessary and sufficient conditions of the existence of the eigenvalues and
construct their asymptotics expansions. The
asymptotics expansions for the associated eigenfunctions are also obtained.
Examples are given. This is a joint work with R. Gadyl'shin.