Put $G= \mathbb{C} \rtimes_{\phi} \mathbb{C} \setminus \{0\}$ where $\phi_{a} (z)= az$ for $a \in \mathbb{C} \setminus \{0\}$. $G$ is a real $4$ dimensional Lie group; then it has a unique left invariant metric which restricts to the standard Euclidean metric at the neutral element. Let $I(G)$ be the group of isometries of $G$ with respect to this invariant metric.

On the other hand $G$ is an open subset of the complex Euclidean space. The group of holomorphic automorphisms of $G$ is denoted by $Aut(G)$.

We constructed $G$ as an obvious generalization of the $2$ dimensional Lie group $H= \mathbb{R} \rtimes \mathbb{R} \setminus \{0\}$, the Poincare upper half plane. We know that the group of holomorphic automorphisms of $H$ preserves the left invariant metric of $H$.

Now a natural question is: What is the structure of $Aut(G)$? What geometric structure on $G$ is preserved by $Aut(G)$?

Are there any relations between $I(G)$ and $Aut(G)$?

Is there a precise description for these two groups?

Considering $G$ as an open set in $\mathbb{C}^2$, is it true to say that all geodesics are either closed or perpendecular to the boundary? (Perpendecular with respect to the usual geometry of $\mathbb{C}^2 = \mathbb{R}^4$?