Normal Forms of the ODE System

In a neighborhood of a stationary point we consider the n-dimensional autonomous formal system of ordinary differential equations. We give 5 types of normal forms (resonant, secondary, Belitskii, complete Belitskii, Gaeta), to which the system is reduced by formal invertible changes of coordinates. We discuss structures, properties and mutual connections of the normal forms. We show that one of them (Gaeta) is wrong.