  The relative equilibria of a symmetric Hamiltonian dynamical system are the
critical points of the so-called augmented Hamiltonian. The underlying
geometric structure of the system is used to decompose the critical point
equations and construct a collection of implicitly defined functions and
reduced equations describing the set of relative equilibria in a neighborhood
of a given relative equilibrium. The structure of the reduced equations is
studied in a few relevant situations. In particular, a persistence result of
Lerman and Singer [LS98] is generalized to the framework of Abelian proper
actions. Also, a Hamiltonian version of the Equivariant Branching Lemma and a
study of bifurcations with maximal isotropy are presented. An elementary
example is presented to illustrate the use of this approach.
