We derive precise asymptotics for eigenvalues for selfadjoint $h$--pseudodifferential operators in dimension two that arise as perturbations of selfadjoint operators with periodic classical flow. When the strength $\epsilon$ of the perturbation is $\ll h$, the spectrum displays a cluster structure, and assuming that $\epsilon \gg h^2$, we may obtain a complete asymptotic description of individual eigenvalues inside subclusters corresponding to regular values of the leading symbol of the perturbation, averaged along the flow. This is joint work with Michael Hitrik and Johannes Sjöstrand.