  Let $G$ be a reductive complex Lie group acting holomorphically on Stein
manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the
quotient mappings. When is there an equivariant biholomorphism of $X$ and $Y$?
A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are
biholomorphic and that the biholomorphism $\phi$ sends the Luna strata of $Q_X$
isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $\phi$. We
demonstrate two homotopy principles in this situation. The first result says
that if there is a $G$-diffeomorphism $\Phi\colon X\to Y$, inducing $\phi$,
which is $G$-biholomorphic on the reduced fibres of the quotient mappings, then
$\Phi$ is homotopic, through $G$-diffeomorphisms satisfying the same
conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. The second
result roughly says that if we have a $G$-homeomorphism $\Phi\colon X\to Y$
which induces a continuous family of $G$-equivariant biholomorphisms of the
fibres $p_X^{-1}(q)$ and $p_Y^{-1}(\phi(q))$ for $q\in Q_X$ and if $X$
satisfies an auxiliary property (which holds for most $X$), then $\Phi$ is
homotopic, through $G$-homeomorphisms satisfying the same conditions, to a
$G$-equivariant biholomorphism from $X$ to $Y$. Our results improve upon
earlier work of the authors and use new ideas and techniques.
