  We consider free and proper cotangent-lifted symmetries of Hamiltonian
systems. For the special case of G = SO(3), we construct symplectic slice
coordinates around an arbitrary point. We thus obtain a parametrisation of the
phase space suitable for the study of dynamics near relative equilibria, in
particular for the Birkhoff-Poincare normal form method. For a general symmetry
group, we observe that for the calculation of the truncated normal forms, one
does not need an explicit coordinate transformation but only its higher
derivatives at the relative equilibrium. We outline an iterative scheme using
these derivatives for the computation of truncated Birkhoff-Poincare normal
forms.
