  Let $D$ and $G$ be copies of the open unit disc in $\C,$ let
  $A$ (resp. $B$) be a measurable subset of $\partial D$ (resp.
  $\partial G$), let $W$ be the 2-fold cross $\big((D\cup A)\times B\big)\cup
\big(A\times(B\cup G)\big),$ and let $M$ be a relatively closed subset of $W.$
Suppose in addition that $A$ and $B$ are of positive one-dimensional Lebesgue
measure and that $M$ is fiberwise polar (resp. fiberwise discrete) and that
$M\cap (A\times B)=\varnothing.$ We determine the "envelope of holomorphy"
$\hat{W\setminus M}$ of $W\setminus M$ in the sense that any function locally
bounded on $W\setminus M,$ measurable on $A\times B,$ and separately
holomorphic on $\big((A\times G) \cup (D\times B)\big)\setminus M$ "extends" to
a function holomorphic on $\hat{W\setminus M}.$
