  In this paper, we report several large classes of affine varieties (over an
arbitrary field $K$ of characteristic 0) with the following property: each
variety in these classes has an isomorphic copy such that the corresponding
isomorphism cannot be extended to an automorphism of the ambient affine space
$K^n$. This implies, in particular, that each of these varieties has at least
two inequivalent embeddings in $K^n$.
  The following application of our results seems interesting: we show that
lines in $K^2$ are distinguished among irreducible algebraic retracts by the
property of having a unique embedding in $K^2$.
