I have the polynomial $f = x^3 - y^2$, considered as a convergent power series $f \in \mathbb C\{x,y\}$. Is there a strategy to compute the group $G$ of right-equivalences (coordinate transformations) under which $f$ is fixed? Given $\varphi \in G$, set $\xi = \varphi(x), \zeta = \varphi(y)$, so that $\xi^3 - \zeta^2 = x^3 - y^2,$ or equivalently $$\tag{1}\label{eq1}(\xi -x)(\xi-\omega x)(\xi - \omega^2x) = (\zeta-y)(\zeta+y),$$ for $\omega = e^{2\pi i / 3}$. Evidently $$\xi = \omega^k x \quad \text{and} \quad \zeta = (-1)^\ell y$$ are valid choices, which gives a subgroup $\mathbb Z/6 \subset G$. But I was wondering if there are any other possibilities?
From \eqref{eq1} I can deduce the following: Each factor on the left hand side vanishes in $(0,0)$, and so the right hand side is contained in $\mathfrak m^3 \subset \mathbb C\{x,y\}$. Hence either $\zeta-y$ or $\zeta+y$ is contained in $\mathfrak m^2$, which implies $$\zeta = \pm y + (\text{terms of order} \geq 2).$$ However, I was not able to deduce a similar thing about $\xi$. Trying to spell everything out in coefficients of power series became a mess quite quickly (though I might be to blame for this one...)