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 Kn. This implies, in particular, that each of these varieties has at least two inequivalent embeddings in Kn. The following application of our results seems interesting: we show that lines in K2 are distinguished among irreducible algebraic retracts by the property of having a unique embedding in K2.