Essential points of conformal vector fields

An essential point of a conformal vector field ξ on a conformal manifold (M,c) is a point around which the local flow of ξ preserves no metric in the conformal class c. It is well-known that a conformal vector field vanishes at each essential point. In this note we show that essential points are isolated. This is a generalization to higher dimensions of the fact that the zeros of a holomorphic function are isolated. As an application, we show that every connected component of the zero set of a conformal vector field is totally umbilical.