  For a H\'enon map of the form $H(x, y) = (y, p(y) - ax)$, where $p$ is a
polynomial of degree at least two and $a \not= 0$, it is known that the
sub-level sets of the Green's function $G^+_H$ associated with $H$ are Short
$\mathbb C^2$'s. For a given $c > 0$, we study the holomorphic automorphism
group of such a Short $\mathbb C^2$, namely $\Omega_c = \{ G^+_H < c \}$. The
unbounded domain $\Omega_c \subset \mathbb C^2$ is known to have smooth real
analytic Levi-flat boundary. Despite the fact that $\Omega_c$ admits an
exhaustion by biholomorphic images of the unit ball, it turns out that its
automorphism group, Aut$(\Omega_c)$ cannot be too large. On the other hand,
examples are provided to show that these automorphism groups are non-trivial in
general. We also obtain necessary and sufficient conditions for such a pair of
Short $\mathbb C^2$'s to be biholomorphic.
