The formal linearization of a semisimple Lie algebra of vector fields about a singular point

Abstract:

A classical theorem by Poincaré gives conditions that a nonlinear ordinary differential equation \[ dx/dt = A(x),\] with $A(0) = 0$ in n variables $x = ({x_1}, \ldots ,{x_n})$ can be reduced to a linear form \[ \frac {{dx’}}{{dt}} = \frac {{\partial A}}{{\partial x}}(0)x’\] by a change of variables $x’ = f(x)$. A generalization is given for a finite set of such differential equations, which form a semisimple Lie algebra.