Theorem 1 (converse of the equality case): $|\mathrm{Sym}^+(C_{m,\alpha})|=2m$ for $m\ge3$ and every nonreal $\alpha$

Let $m\ge3$ be an integer and let $\alpha\in\mathbb C$ with $\alpha\notin\mathbb R$; no condition is imposed on $|\alpha|$. Let $C_{m,\alpha}=\{z\in\mathbb C:\operatorname{Re}\bigl(z^m(|z|^2+\alpha)\bigr)=0\}$ as in equation (1), and let $\mathrm{Sym}^+(C_{m,\alpha})$ be the group of orientation-preserving Euclidean isometries $z\mapsto az+b$, $|a|=1$, mapping $C_{m,\alpha}$ onto itself.

Then there is a polynomial $f\in\mathbb R[x,y]$ with the following properties:

1. $f$ is irreducible in $\mathbb C[x,y]$ and has total degree $m+2$;
2. the real zero set of $f$ is $C_{m,\alpha}$, that is, $\{z\in\mathbb C: f(\operatorname{Re}z,\operatorname{Im}z)=0\}=C_{m,\alpha}$;
3. $C_{m,\alpha}$ is infinite;
4. $C_{m,\alpha}$ is not a circle: there are no $c\in\mathbb C$ and real $R>0$ with $C_{m,\alpha}=\{z:|z-c|=R\}$;
5. the group $\mathrm{Sym}^+(C_{m,\alpha})$ has order $2m=2\deg f-4$:

This is the converse in Theorem 1: every curve of the family (1) with $m\ge3$ satisfies the hypotheses of that theorem in degree $d=m+2$ and attains the rotation bound $2d-4$. The statement requires only $\alpha\notin\mathbb R$, while the family in the note also has $|\alpha|=1$.

Formalization Note: the curve is a subset of $\mathbb C$ and $\mathrm{Sym}^+(C_{m,\alpha})$ is a subgroup of the group ℂ ≃ᵢ ℂ of isometric bijections; its order is Nat.card, so with $2m>0$ finiteness is part of the conclusion.