From a factorization formula for the monodromy element of a periodic Lie system on a Lie group, we show that the concepts geometric and dynamical phases are naturally defined for such class of systems. An interpretation of the phases is given in terms of the Poisson
geometry and the Hamiltonian dynamics induced by the coadjoint action on the Lie coalgebra of the group. Applying these results we give a general formulae for the reconstruction dynamics of a system with symmetry which generalize previous results given by Marsden et al. in .