Further remarks on rigidity of H\'{e}non maps

For a H\'{e}non map $H$ in $\mathbb{C}^2$, we characterize the polynomial automorphisms of $\mathbb{C}^2$ which keep any fixed level set of the Green function of $H$ completely invariant. The interior of any non-zero sublevel set of the Green function of a H\'{e}non map turns out to be a Short $\mathbb{C}^2$ and as a consequence of our characterization, it follows that there exists no polynomial automorphism apart from possibly the affine automorphisms which acts as an automorphism on any of these Short $\mathbb{C}^2$'s. Further, we prove that if any two level sets of the Green functions of a pair of H\'{e}non maps coincide, then they almost commute.