  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.
