In my research I encountered automorphisms of the ring of formal power series $$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$ which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm wondering about the structure of such a $\varphi$. Apart from the obvious automorphisms $x \mapsto e^{\frac{2 \pi i}{3}}x$ and $y \mapsto -y$, how can such a $\varphi$ look?

I recently learned about the following way to produce such $\varphi$: Consider the vector field $$H = 2y \frac{\partial}{\partial x} + 3x^2 \frac{\partial}{\partial y}.$$ Clearly, $H(f) = 0$. So any flow of $H$ preserves $f$, so taking the flow at some fixed time $t$ defines a desired $\varphi$. Also, multiplying $H$ with any power series $g$ has the same property. It is also clear that any vector field $X$ with $X(f) = 0$ is of the form $gH$ for some $g$.

My professor said that those might in fact be all possible automorphisms, but he wasn't sure about a reference. Where could I find something in that direction?

I also asked a similar question here, but I figured that the answer might not be so simple, and this question is a bit more focused.