ABSTRACT
This chapter discusses the local properties of holomorphic vector fields on complex manifolds, in particular of some local properties that amount to integrability conditions for holomorphic systems of ordinary differential equations. They are an important application of the theory of holomorphic functions of several variables, albeit an application of only relatively elementary aspects of that theory, and so for the sake of completeness. Systems of differential equations arise when considering the problem of simplifying a germ of a holomorphic vector field in Cn by an appropriate local change of coordinates. With the structure thus introduced on the tangent variety it is possible to introduce the following concept, a natural extension of the corresponding notion for complex manifolds.
