Functions holomorphic along holomorphic vector fields
  The main result of the paper is the following generalization of Forelli's theorem: Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with the eigenvalues whose ratios are positive reals. Then any function $\phi$ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p.   We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary. 
Kang-Tae Kim, Evgeny Poletsky and Gerd Schmalz