  This is the first in a series of works devoted to small non-selfadjoint
perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2.
In the present work we treat the case when the classical flow of the
unperturbed part is periodic and the strength $\epsilon$ of the perturbation is
$\gg h$ (or sometimes only $\gg h^2$) and bounded from above by $h^{\delta}$
for some $\delta>0$. We get a complete asymptotic description of all
eigenvalues in certain rectangles $[-1/C, 1/C]+ i\epsilon [F_0-1/C,F_0+1/C]$.
