  We find asymptotical expansions as $\nu \to 0$ for integrals of the form
$\int_{\mathbb{R}^d} F(x) / \big(\omega(x)^2 + \nu^2\big)\, dx$, where
sufficiently smooth functions $F$ and $\omega$ satisfy natural assumptions for
their behaviour at infinity and all critical points of the function $\omega$
from the set $\{\omega(x) = 0\}$ are non-degenerate. These asymptotics play a
crucial role when analysing stochastic models for non-linear waves systems. Our
result generalizes that of [S. Kuksin, Russ. J. Math. Phys.'2017] where a
similar asymptotics was found in a particular case when $\omega$ is a
non-degenerate quadratic form of the signature $(d/2,d/2)$ with even $d$.
