  A Danilov-Gizatullin surface is an affine surface $V$ which is the complement
of an ample section $S$ of a Hirzebruch surface. The remarkable theorem of
Danilov and Gizatullin states that the isomorphism class of $V$ depends only on
the self-intersection number $S^2$. In this paper we apply their theorem to
present $V$ as the quotient of an affine threefold by a torus action, and to
prove that the Lie algebra generated by the complete algebraic vector fields on
$V$ coincides with the set of all algebraic vector fields.
