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

In the paper “On rigid germs of finite morphisms of smooth surfaces” (Sb. Math., 211:10 (2020), 1354–1381), we defined a map $\beta\colon{\mathcal R\to\mathcal{B}el}$ 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 $f\colon\mathbb P^1\,{\to}\,\mathbb P^1$, considered up to the action of $\mathrm{PGL}(2,\mathbb C)$. In this article the inverse images of this map are investigated in terms of monodromies of Belyi pairs.
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