Let $G:=\mathbb C^n/\Gamma$ be an abelian complex Lie group where $\Gamma$ is a discrete subgroup of $\mathbb C^n$. Suppose $G$ is a toroidal group i.e $G$ contains a normal subgroup $H\cong (\mathbb C^*)^m$ s.t $G/H$ is compact. (Or equivalently $G$ has no non-constant holomorphic functions).
Let $V:=span_\mathbb R\ \Gamma$ and $W:=V\cap iV$ be the maximal complex subspace of $V$. Then the orbit of $W$ through the identity of $G$ is isomorphic to $W/(W\cap \Gamma)$ which is closed and dense in the maximal compact subgroup $K:=V/\Gamma$ of $G$.
My questions are: Are all orbits of $W$ in $G$ dense in $K$?
Do the orbits of $W$ form a complex foliation of $K$?