The characterization of holomorphic vector fields vanishing at an infinite type point

This paper consider a rotationally symmetric real hypersurface in C 2 around the origin. We also assume that the origin is a point of infinite type in the sense of DʼAngelo. We prove that a holomorphic vector field defined on a neighborhood of origin which is tangent to the real hypersurface and vanishing at the origin is indeed a vector field generating rotational transformations.