Small families of complex lines for testing holomorphic extendibility

Let B be the open unit ball in C^2 and let a, b be two points in B. It is known that for every positive integer k there is a function f in C^k(bB) which extends holomorphically into B along any complex line passing through either a or b yet f does not extend holomorphically through B. In the paper we show that there is no such function in C^\infty (bB). Moreover, we obtain a fairly complete description of pairs of points a, b in C^2 such that if a function f in C^\infty(bB) extends holomorphically into B along each complex line passing through either a or b that meets B, then f extends holomorphically through B.