  We single out a class of Lagrangians on a group manifold, for which one can
introduce non-canonical coordinates in the phase space, which simplify the
construction of the Poisson structure without explicitly calculating the Dirac
bracket. In the case of $SO(3)$\,- manifold, the application of this formalism
leads to the Poincar\'e-Chetaev equations. The general solution to these
equations is written in terms of exponential of the Hamiltonian vector field.
