Bifurcations in Hamiltonian systems with a reflecting symmetry

A reflecting symmetry q →−q of a Hamiltonian system does not leave the symplectic structure dq ∧ d p invariant and is therefore usually associated with a reversible Hamiltonian system. However, if q → −q leads to H → −H, then the equations of motion are invariant under the reflection. Such a symmetry imposes strong restrictions on equilibria with q = 0. We study the possible bifurcations triggered by a zero eigenvalue and describe the simplest bifurcation triggered by non-zero eigenvalues on the imaginary axis.