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Let $f:S^2\to \mathbb{R}$ be a Morse function on the $2$-sphere and $K$ be aconnected component of some level set of $f$ containing at least one saddlecritical point. Then $K$ is a $1$-dimensional CW-complex cellularly embeddedinto $S^2$, so the complement $S^2\setminus K$ is a union of open $2$-disks$D_1,\ldots, D_k$. Let $\mathcal{S}_{K}(f)$ be the group of isotopic to theidentity diffeomorphisms of $S^2$ leaving invariant $K$ and also each level set$f^{-1}(c)$, $c\in\mathbb{R}$. Then each $h\in \mathcal{S}_{K}(f)$ induces acertain permutation $\sigma_{h}$ of those disks. Denote by $G = \{ \sigma_h\mid h \in \mathcal{S}_{K}(f)\}$ be the group of all such permutations. Weprove that $G$ is isomorphic to a finite subgroup of $SO(3)$.

Automorphisms of cellular divisions of $2$-sphere induced by functions with isolated critical points