  We study semiclassical asymptotics for spectra of non-selfadjoint
perturbations of selfadjoint analytic $h$-pseudodifferential operators in
dimension 2, assuming that the classical flow of the unperturbed part is
completely integrable. Complete asymptotic expansions are established for all
individual eigenvalues in suitable regions of the complex spectral plane, near
the edges of the spectral band, coming from rational flow-invariant Lagrangian
tori.
