Normal Forms and Unfoldings of Linear Systems in Eigenspaces of (Anti)-Automorphisms of Order Two

In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and Hamiltonian, which have both time-preserving and time-reversing symmetries. However the theory gives a uniform method to obtain normal forms and unfoldings for a wide variety of linear differential equations with additional structure. We give several examples and include a discussion of the phenomenon of orbit splitting. As a consequence of orbit splitting we observe passing and splitting of eigenvalues in unfoldings.