  Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a
holomorphic foliation on $M$ invariant under a transitive group $G$ of
biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a
holomorphic $G$-equivariant submersion $\pi \colon M \to N$ onto a
$G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an
automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in
$\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a
connected complex submanifold of $D$.
