An anti-Möbius map sending $\widehat C_{m,\alpha}$ into $\widehat C_{m,\beta}$ carries $V_\alpha$ onto $V_\beta$

Let $m\ge1$ be an integer and let $\alpha,\beta\in\mathbb C$ with $\alpha\ne\bar\alpha$ and $\beta\ne\bar\beta$; no condition on their moduli is imposed. For $\gamma\in\{\alpha,\beta\}$ let $\widehat C_{m,\gamma}$ be the closure in the Riemann sphere $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ of $C_{m,\gamma}=\{z\in\mathbb C:\operatorname{Re}\bigl(z^m(|z|^2+\gamma)\bigr)=0\}$; complex conjugation $p\mapsto\bar p$ acts on $\widehat{\mathbb C}$, with $\bar\infty=\infty$. For $x=(x_0,x_1)$ and $y=(y_0,y_1)$ in $\mathbb C^2$ let

the bihomogenization of $P_\gamma(X,Y)=X^m(\gamma+XY)+Y^m(\bar\gamma+XY)$, and let $V_\gamma=\{([x],[y])\in\mathbb P^1\times\mathbb P^1:F_\gamma(x,y)=0\}$.

Let $g\in\mathrm{GL}_2(\mathbb C)$ with rows $(a,b)$ and $(c,d)$ be any invertible matrix, $M_g(z)=\frac{az+b}{cz+d}$ the Möbius transformation of $\widehat{\mathbb C}$ it defines, and $\bar g$ the matrix with conjugated entries; $\mathrm{GL}_2(\mathbb C)$ acts on $\mathbb P^1$ by $h\,[x]=[hx]$. Assume that the anti-Möbius map $p\mapsto M_g(\bar p)$ sends $\widehat C_{m,\alpha}$ into $\widehat C_{m,\beta}$, that is, $M_g(\bar p)\in\widehat C_{m,\beta}$ for every $p\in\widehat C_{m,\alpha}$. Then for every $(p_1,p_2)\in\mathbb P^1\times\mathbb P^1$

In the proof of Theorem 2 an anti-Möbius map $M\circ\mathrm{conj}$ acts on the complexification by $(X,Y)\mapsto(M(Y),\overline M(X))$: complex conjugation on the sphere becomes the exchange of the two factors of $\mathbb P^1\times\mathbb P^1$. This is the antiholomorphic case of the step showing that an equivalence of the spherical real loci carries $V_\alpha$ onto $V_\beta$; only an inclusion is assumed.

Formalization Note: the anti-Möbius map is $p\mapsto g\cdot\bar p$, with OnePoint.map star as the conjugation of OnePoint ℂ and an arbitrary $g$ in GL (Fin 2) ℂ; membership in $V_\gamma$ is tested on chosen representatives.