Periodic solutions of symmetric autonomous Newtonian systems

In this paper we show the existence and bifurcation of T-periodic solutions of a special form for an autonomous Newtonian system with symmetry. If the phase-space is equipped with the structure of an orthogonal representation and the potential is invariant, then for every such a solution the set of indices of nonvanishing Fourier coefficients is finite and depends on only. If the potential V depends on the squares of complex coordinates, then for every such a solution T is the minimal period.