On continuation of functions with polar singularities

The main result is
Theorem 1 . {\it If $f$ is a holomorphic function on the polydisk $'U\times U_n$ in $\mathbf C^n,$ and for each fixed $'a$ in some nonpluripolar set $E\subset{}'U$ the function $f('a,z_n)$ can be continued holomorphically to the whole plane with the exception of some polar set of singularities, then $f$ can be continued holomorphically to $('U\times\mathbf C)\setminus S,$ where $S$ is a closed pluripolar subset of $'U\times\mathbf C$.}
Some generalizations are also given, along with corollaries on extension of functions with analytic sets of singularities.
Bibliography: 13 titles.