  We study the action of the automorphism group of the $2$ complex dimensional
manifold symmetrized bidisc $\mathbb{G}$ on itself. The automorphism group is 3
real dimensional. It foliates $\mathbb{G}$ into leaves all of which are 3 real
dimensional hypersurfaces except one, viz., the royal variety. This leads us to
investigate Isaev's classification of all Kobayashi-hyperbolic 2 complex
dimensional manifolds for which the group of holomorphic automorphisms has real
dimension 3 studied by Isaev. Indeed, we produce a biholomorphism between the
symmetrized bidisc and the domain
  \[\{(z_1,z_2)\in \mathbb{C} ^2 : 1+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|,
Im(z_1 (1+\overline{z_2}))>0\}\] in Isaev's list. Isaev calls it $\mathcal
D_1$. The road to the biholomorphism is paved with various geometric insights
about $\mathbb{G}$. Several consequences of the biholomorphism follow including
two new characterizations of the symmetrized bidisc and several new
characterizations of $\mathcal D_1$. Among the results on $\mathcal D_1$, of
particular interest is the fact that $\mathcal D_1$ is a "symmetrization". When
we symmetrize (appropriately defined in the context in the last section) either
$\Omega_1$ or $\mathcal{D}^{(2)}_1$ (Isaev's notation), we get $\mathcal D_1$.
These two domains $\Omega_1$ and $\mathcal{D}^{(2)}_1$ are in Isaev's list and
he mentioned that these are biholomorphic to $\mathbb{D} \times \mathbb{D}$. We
produce explicit biholomorphisms between these domains and $\mathbb{D} \times
\mathbb{D}$.
