Reduction of T*SE(3) to the Poisson structure for a symmetric top

Symmetrical top is a special case of a general top. The canonical Poisson structure on T*SE(3) is the common method of its description. This Poisson structure is invariant under the right action of SO(3). However the Hamiltonian of the symmetrical top is only invariant under the right action of the subgroup S1 that corresponds to the rotation around the top symmetry axis. Then the Poisson structure for a symmetric top is obtained by reduction T*SE(3)/S1 is shown.