  Let $B^n$ be the $n$-dimensional unit complex ball and let $a$ and $b$ be two
distinct points in its closure. Let $f$ be a real-analytic function on the
complex unit sphere $\partial B^n.$ Suppose that for any complex line $L,$
meeting the two points set $\{a,b\},$ the function $f$ admits one-dimensional
holomorphic extension in the cross-section $L \cap B^n.$ Then $f$ is the
boundary value of a function holomorphic in $B^n$. Two points can not be
replaced by a single point. The proof essentially uses recent result of the
author about characterization of polyanalytic functions in the complex plane.
