Asymptotics of two-dimensional integrals depending singularity on a small parameter

The asymptotics is constructed for integrals of form $\iint\limits_w \frac{dxdy}{\epsilon^2+\chi(x,y)}$ where $\omega$ is some vicinity of a critical point $(0,0)$ in which function $\chi(x,y)$ is equal to zero. It is considered the case in which function $\chi(x,y)$ addresses in a zero on two crossed curves and has a special appearance.