Automorphisms of affine surfaces. I

We study the group of automorphisms $\operatorname{Aut}(X)$ of an affine surface $X$ which can be made complete by adding a zigzag. This study is based on the computation of the action of $\operatorname{Aut}(X)$ on a certain tree $\Delta_X$ associated with the surface $X$. Our results are used to give a description of forms of the surface $X$ and of algebraic subgroups of $\operatorname{Aut}(X)$.
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