Uniqueness of $\bf C^*$- and $\bf C_+$-actions on Gizatullin surfaces

A Gizatullin surface is a normal affine surface $V$ over $\bf C$, 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 $\bf C^*$-actions and $\bf A^1$-fibrations on such a surface $V$ up to automorphisms. The latter fibrations are in one to one correspondence with $\bf C_+$-actions on $V$ considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one $\bf A^1$-fibration $V\to S$ up to an isomorphism of the base $S$. Moreover an effective $\bf C^{*}$-action on them, if it does exist, is unique up to conjugation and inversion $t\mapsto t^{-1}$ of $\bf C^*$. Obviously uniqueness of $\bf C^*$-actions fails for affine toric surfaces; however we show in this case that there are at most two conjugacy classes of $\bf A^1$-fibrations. There is a further interesting family of non-toric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $\bf C^*$-actions and $\bf A^1$-fibrations. In the present paper we obtain a criterion as to when $\bf A^1$-fibrations of Gizatullin surfaces are conjugate up to an automorphism of $V$ and the base $S$. We exhibit as well a large subclasses of Gizatullin $\bf C^{*}$-surfaces for which a $\bf C^*$-action is essentially unique and for which there are at most two conjugacy classes of $\bf A^1$-fibrations over $\bf A^1$.