  We study the actions of a Lie group $G$ by birationally extendible
automorphisms on a domain $D\subset C^n$. For a large class of such domains
defined by polynomial inequalities, all automorphisms are of this type. In the
cases 1) $G$ has finitely many components or 2) the degree of the automorphisms
is bounded, we prove that the action of $G$ is projectively linearizable, i.e.
there exist a linear representation of $G$ on some $ C^{N+1}$ and a holomorphic
$G$-equivariant embedding $i: D\to P^N$, which is a restriction of a rational
mapping. As a corollary we obtain as many rational invariant functions as the
dimension of generic orbits allows. A hard copy is available from
<EMAIL_ADDRESS>