The Möbius symmetry group of $\widehat{C}_{m,\alpha}$ has exactly $4m$ elements

Let $m\ge 2$ be an integer, let $\alpha\in\mathbb{C}$ satisfy $|\alpha|=1$ and $\alpha\ne\bar\alpha$ (so $\alpha$ is not real), and let $C_{m,\alpha}=\{z\in\mathbb{C} : \operatorname{Re}\bigl(z^m(|z|^2+\alpha)\bigr)=0\}$. Let $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ be the Riemann sphere and let $\widehat{C}_{m,\alpha}$ be the closure of $C_{m,\alpha}$ in $\widehat{\mathbb{C}}$. A matrix $h=(h_{jk})\in\mathrm{GL}_2(\mathbb{C})$ acts on $\widehat{\mathbb{C}}$ by the Möbius transformation $p\mapsto(h_{11}p+h_{12})/(h_{21}p+h_{22})$, with the usual conventions at $\infty$ and at the pole. Let $G_{m,\alpha}$ be the group, under composition, of the bijections $\sigma$ of $\widehat{\mathbb{C}}$ that are induced by some matrix in $\mathrm{GL}_2(\mathbb{C})$ and satisfy $\sigma(p)\in\widehat{C}_{m,\alpha}\iff p\in\widehat{C}_{m,\alpha}$ for every $p\in\widehat{\mathbb{C}}$; these are the holomorphic Möbius transformations mapping $\widehat{C}_{m,\alpha}$ onto itself.

Then $G_{m,\alpha}$ is finite, with

This is the order of the ambient symmetry group in Theorem 2 of the note, whose elements are the maps of equation (3): $z\mapsto cz$ with $c^{2m}=1$, and $z\mapsto c/z$ with $c^{2m}=\bar\alpha^{2}$. Anti-Möbius maps are not elements of $G_{m,\alpha}$; that none of them preserves $\widehat{C}_{m,\alpha}$ is a separate result. The quartic case $m=2$ is included.

Formalization Note: The sphere is OnePoint ℂ. The group $G_{m,\alpha}$ is a subgroup of the permutations of the sphere, namely the image of $\mathrm{GL}_2(\mathbb{C})$ intersected with the stabilizer of $\widehat{C}_{m,\alpha}$, so proportional matrices give the same element; $\widehat{C}_{m,\alpha}$ is the topological closure of the image of $C_{m,\alpha}$. The cardinality is Nat.card, so the equation also asserts finiteness.