Complexifications of local actions and the Bergman metric

Let $\,G=({\Bbb R},+)\,$ act by biholomorphisms on a Stein manifold $\,X\,$ which admits the Bergman metric. We show that $\,X\,$ can be regarded as a $\,G$-invariant domain in a ``universal" complex manifold $\,X^*\,$ on which the complexification $\,({\Bbb C},+)\,$ of $\,G\,$ acts. The analogous result holds for actions of a larger class of real Lie groups containing, e.g., abelian and certain nilpotent ones. For holomorphic actions of such groups on Stein manifolds, necessary and sufficient conditions for the existence of $\,X^*\,$ are given.