Normal Forms for Equivariant Differential Equations

We show that if the equation x′=A(t)x+f(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'=A(t) x + f(t,x)$$\end{document} is equivariant (respectively, reversible), then any normal form as well as the coordinate change taking the original equation to the normal form have equivariance (respectively, reversibility) properties. The proof depends on writing down somewhat explicit coordinate changes based on the block diagonalization of the linear part so that each block corresponds to a connected component of the nonuniform spectrum.