On the Holomorphic Automorphism Group of Danielewski surfaces

We will quickly go through some famous result in the area of the tame generator conjecture about the algebraic and the holomorphic automorphism group of \C2. Then we will give a version how to reformulate those statements for a modified version of \C2, namely the surfaces given by {xy=p(z)} also referred to as Danielewski surfaces. These questions are related to the study of one parameter subgroups, it turned out that there are some interesting phenomena. In particular the Lie algebra generated by locally nilpotent derivations is not dense in the Lie algebra of volume preserving vector fields, whereas it is dense for \C2.