  In this paper we give a description of hypersurfaces with trivial ring
$AK(S)$, introduced by the second author as following. Let $X$ be an affine
variety and let $G(X)$ be the group generated by all $\Bbb {C}^+$-actions on
$X$. Then $AK(X)$ is the subring of all regular $G(X)-$ invariant functions on
$X.$ We show that a smooth affine surface $S$ with $AK(S)=\Bbb C$ is
quasihomogeneous and so may be obtained from a smooth rational projective
surface by deleting a divisor of special form, which is called a ``zigzag''. We
denote by $A$ the set of all such surfaces, and by $H$ those which have only
three components in the zigzag. We prove that for a surface $S \in A$ the
following statements are equivalent:
  1. $S$ is isomorphic to a hypersurface;
  2. $S$ is isomorphic to a hypersurface, defined by equation $xy=p(z)$ in
$\Bbb {C}^3 ,$ where $p$ is a polynomial with simple roots only;
  3. $S$ admits a fixed-point free $\Bbb {C}^+$- action;
  4. $S\in H.$ Moreover, if $S_1 $ belongs to $H,$ and $S_2$ does not, then
$S_1\times \Bbb {C}^k\not\cong S_2\times \Bbb {C}^k$ for any $k\in\Bbb N$.
