  We construct holomorphic families of proper holomorphic embeddings of $\C^k$
into $\C^n$ ($0<k<n-1$), so that for any two different parameters in the family
no holomorphic automorphism of $\C^n$ can map the image of the corresponding
two embeddings onto each other. As an application to the study of the group of
holomorphic automorphisms of $\C^n$ we derive the existence of families of
holomorphic $\C^*$-actions on $\C^n$ ($n\ge 5$) so that different actions in
the family are not conjugate. This result is surprising in view of the long
standing Holomorphic Linearization Problem, which in particular asked whether
there would be more than one conjugacy class of $\C^*$ actions on $\C^n$ (with
prescribed linear part at a fixed point).
