Transitivity of Automorphism Groups of Gizatullin Surfaces

We show that the automorphism group of a certain subclass of smooth Gizatullin surfaces with a distinguished and rigid extended divisor is generated by automorphisms of |$\mathbb A^1$|-fibrations. Moreover, such surfaces yield examples of smooth Gizatullin surfaces with a nontransitive action of the automorphism group. Thus, they represent counter-examples to Gizatullin's conjecture. For such surfaces, we give an explicit orbit decomposition of the natural action of the automorphism group in some special cases. It turns out that the automorphism groups of such surfaces are amalgamated products of two subgroups.