  This is the second in a series of works devoted to small non-selfadjoint
perturbations of selfadjoint semiclassical pseudodifferential operators in
dimension 2. As in our previous work, we consider the case when the classical
flow of the unperturbed part is periodic. Under the assumption that the flow
average of the leading perturbation vanishes identically, we show how to obtain
a complete asymptotic description of the individual eigenvalues in certain
domains in the complex plane, provided that the strength of the perturbation
$\epsilon$ is $\gg h^{1/2}$, or sometimes only $\gg h$, and enjoys the upper
bound $\epsilon={\cal O}(h^{\delta})$, for some $\delta>0$.
