I have the following question, which I'm sure must be explored somewhere.

Consider a group of polynomial automorphisms of $\mathbb{A}^2_\mathbb{C}$ preserving a standard hermitian metric. Is there any description of this group?

I know that analogous question for symplectomorphisms has been studied, for example, here the authors prove infinite transitivity of the group of polynomial symplectomorphisms.

Still, I was unable to find anything similar for the group of polynomial isometries.