The authors study the behavior near an isolated singularity at the origin of an analytic vector field \(X\) on the plane for which the linear part \(DX(0)\) has both eigenvalues zero, but is not itself zero (a “nilpotent singularity”). They give a new proof of a theorem of Andreev that characterizes which of such critical points are monodromic, meaning that there is a first return map defined on a section of the flow of the vector field with one endpoint at the origin. After introducing a special form for vector fields having a monodromic nilpotent singularity at the origin, they characterize, for several specialized cases, those singularities of this type that are in fact centers. Techniques used include index theory and the generalized trigonometric functions of Lyapunov.