  Following an approach of Dolgachev, Pinkham and Demazure, we classified in
math.AG/0210153 normal affine surfaces with hyperbolic $\C^{*}$-actions in
terms of pairs of $\Q$-divisors $(D_+,D_-)$ on a smooth affine curve. In the
present paper we show how to obtain from this description a natural equivariant
completion of these $\C^*$-surfaces. Using elementary transformations we deduce
also natural completions for which the boundary divisor is a standard graph in
the sense of math.AG/0511063 and show in certain cases their uniqueness. This
description is especially precise in the case of normal affine surfaces
completable by a zigzag i.e., by a linear chain of smooth rational curves. As
an application we classify all zigzags that appear as boundaries of smooth or
normal $\C^*$-surfaces.
