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We study the distribution of eigenvalues for non-selfadjoint perturbations ofselfadjoint semiclassical analytic pseudodifferential operators in dimensiontwo, assuming that the classical flow of the unperturbed part is completelyintegrable. An asymptotic formula of Weyl type for the number of eigenvalues ina spectral band, bounded from above and from below by levels corresponding toDiophantine invariant Lagrangian tori, is established. The Weyl law is given interms of the long time averages of the leading non-selfadjoint perturbationalong the classical flow of the unperturbed part.

Diophantine tori and Weyl laws for non-selfadjoint operators in dimension two