Periodic solutions of a many-rotator problem in the plane

We consider the many-rotator system whose motions in the plane are characterized by the Newtonian equations of motion where superimposed arrows denote three-vectors living in the plane in which all motions take place, is a unit vector orthogonal to that plane, the symbol denotes the usual three-dimensional vector product and ω, αnm, α'nm are 2N(N-1) + 1 arbitrary real constants (without loss of generality ω>0). This model is invariant under rotations and translations (in the plane); it is Hamiltonian provided αmn = αnm,α'mn = α'nm; it is not known to be integrable (for N>2), unless all the coupling constants α'nm vanish (α'nm = 0) and the coupling constants αnm either also all vanish (αnm = 0; trivial case, all motions completely periodic, with period T = 2π/ω) or are all equal to unity (αnm = 1; integrable/solvable case, all motions completely periodic, with period at most T' = TN!). We prove that this model generally possesses a large class of solutions (corresponding to a set of initial conditions containing a non-empty open set) which are completely periodic with period T = 2π/ω. Analogous results also hold for more general evolution equations, interpretable as appropriately (linearly) deformed versions of those characterizing geodesic motions in N-dimensional space.