Rigid germs of finite morphisms of smooth surfaces and rational Belyi pairs

In \cite{K-rig}, a map $\beta :\mathcal{R}\to \mathcal{B}el$ from the set $\mathcal{R}$ of equivalence classes of rigid germs of finite morphisms branched in germs of curves having $ADE$ singularity types onto the set $\mathcal{B}el$ of rational Belyi pairs $f:{\mathbb{P}}^1\to {\mathbb{P}}^1$ considered up to the action of $PGL(2,\mathbb{C})$ was defined. In this article, the inverse images of this map are investigated in terms of monodromies of Belyi pairs.