  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.
