An isometry of $\mathbb{C}$ preserves $C_{m,\alpha}$ if and only if it is a rotation $z\mapsto cz$ with $c^{2m}=1$

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An isometry of $\mathbb{C}$ preserves $C_{m,\alpha}$ if and only if it is a rotation $z\mapsto cz$ with $c^{2m}=1$ · Prove2Me