Some problems of affine algebraic geometry

Abstract: There are a lot of questions related to affine algebraic  geometry which have very simple formulations but which, sometimes, are  very difficult to answer. For example, are the surfaces given by  $x^ny=z^2-1$ isomorphic for different $n$? or is a hypersurface given  by $x+x^2y+z^2+t^3=0$ isomorphic to three-dimensional affine space?
In my talk I'll answer these and some other questions with the help of  automorphism groups of these objects.