Let $R_n$ be the integral polynomial ring $\mathbb{Z}[x_1,x_2,...,x_n]$, let $A_n$ be the group of ring automorphisms $\mathrm{Aut}(R_n)$, and for $f\in R_n$ let $\mathrm{Aut}(f)=\{\alpha\in A_n\ |\ \alpha(f)=f\}$.

Define a polynomial $f\in R_n$ to be *interesting* if $\deg_{x_i}(f)\geq 1$ for $1\leq i\leq n$ and there is an infinite order element in $\mathrm{Aut}(f)$.

Since $A_1=\{n\pm x\ |\ n\in \mathbb{Z}\}$ there do not appear to be any interesting members of $R_1$.

On the other hand, $\kappa=x^2+y^2+z^2-xyz-2$ is interesting since Horowitz showed in *Induced automorphisms on Fricke characters of free groups* that $\mathrm{Aut}(\kappa)\cong \mathrm{PGL}(2, \mathbb{Z}) \rtimes (\mathbb{Z}/2 \oplus \mathbb{Z}/2)$.

Hence, there are interesting members of $R_n$ for all $n\geq 3$.

Here is my question:

Are there any interesting planar polynomials?

Precisely, do there exist $f\in \mathbb{Z}[x,y]$ with degree in $x$ and $y$ at least 1 so that $\mathrm{Aut}(f)$ contains an infinite order element?

I imagine that the structure of the affine Cremona group will be relevant here. See *Two-dimensional Cremona groups acting on simplicial complexes* by Wright for a structure theorem relevant to $A_2$ (Theorem 2.4 with $k=\mathbb{Q}$).

**Update**: Given the helpful comments by Yves, which give an answer to the original post (and even the slightly edited version), I have a second question to ask that is too related to post separately.

We now say $f\in R_n$ is *very interesting* if it is interesting and $f\not=\alpha(g)$ where $\alpha\in A_n$ and $g$ has degree 0 in some variable. I believe $\kappa$ remains an example (although it does not seem obvious to me).

Here is the second question:

Are there any very interesting planar polynomials?