Uniqueness of C*- and C+-actions on Gizatullin surfaces

A Gizatullin surface is a normal affine surface V over C, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of C*-actions and A(1)-fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with C+-actions on V considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one A(1)-fibration V -> S up to an isomorphism of the base S. Moreover, an effective C*-action on them, if it does exist, is unique up to conjugation and inversion t -> t(-1) of C*. Obviously, uniqueness of C*-actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov{Gizatullin surfaces, where there are in general several conjugacy classes of C*-actions and A(1)-fibrations, see, e.g., [FKZ(1)]. In the present paper we obtain a criterion as to when A(1)-fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base S congruent to A(1). We exhibit as well large subclasses of Gizatullin C*-surfaces for which a C*-action is essentially unique and for which there are at most two conjugacy classes of A(1)-fibrations over A(1).