Non-extendable isomorphisms between affine varieties

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 $K^2$.