  A characterization of maximal domains of existence of adapted complex
structures for Riemannian homogeneous manifolds under certain extensibility
assumptions on their geodesic flow is given. This is applied to generalized
Heisenberg groups and naturally reductive Riemannian homogeneous spaces. As an
application it is shown that the case of generalized Heisenberg groups yields
examples of maximal domains of definition for the adapted complex structure
which are are neither holomorphically separable, nor holomorphically convex.
