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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 branchedin 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 theaction of $PGL(2,\mathbb C)$ was defined. In this article, the inverse imagesof this map are investigated in terms of monodromies of Belyi pairs.

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