Solution of certain integrable dynamical systems of Ruijs']jsenaars-Schneider type with completely periodic trajectories

The first main result of this paper is the solution of the (complex) equations of motion (z)double over dot(j) + i Omega(z)over dot(j) = Sigma(k=1,k not equal j)(n) (z)over dot(j)(z)over dot(k)f(z(j)-z(k)) with f(z) = 2a cotgh (az)/[1 + r(2)sinh(2)(az)], and the consequent confirmation of the conjecture that all, the trajectories of this dynamical system are completely periodic with period (at most) T' = Tn!, T = 2 pi/Omega. We also discuss a symplectic reduction scheme which features new Lie-theoretic aspects for these systems. These developments are introduced here in the perspective of applying them in future studies to implement geometric quantization techniques.