  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 $\bf A^1$.
