  We study a kind of modification of an affine domain which produces another
affine domain. First appeared in passing in the basic paper of O. Zariski
(1942), it was further considered by E.D. Davis (1967). The first named author
applied its geometric counterpart to construct contractible smooth affine
varieties non-isomorphic to Euclidean spaces. Here we provide certain
conditions which guarantee preservation of the topology under a modification.
  As an application, we show that the group of biregular automorphisms of the
affine hypersurface $X \subset C^{k+2}$ given by the equation
$uv=p(x_1,...,x_k)$ where $p \in C[x_1,...,x_k],$ acts $m-$transitively on the
smooth part reg$X$ of $X$ for any $m \in N.$ We present examples of such
hypersurfaces diffeomorphic to Euclidean spaces.
