Poisson integrator for symmetric rigid bodies

We derive an explicit second order reversible Poisson integrator for symmetric rigid bodies in space (i.e. without a fixed point). The integrator is obtained by applying a splitting method to the Hamiltonian after reduction by the $S^1$ body symmetry. In the particular case of a magnetic top in an axisymmetric magnetic field (i.e. the Levitron) this integrator preserves the two momentum integrals. The method is used to calculate the complicated boundary of stability near a linearly stable relative equilibrium of the Levitron with indefinite Hamiltonian.