Generalizations of the Neumann system

where yr = dxr/ds. The equations Lxr — arxr (r = 0 , . . . , g) are equivalent to the classical Neumann system [7]. H. Flaschka [3] obtained similar results from a different point of view. His approach is based on the articles [2 and 5] of I. V. Cherednik and I. M. Krichever. The familiar Lax pairs, the constants of motion and the quadrics of the Neumann system emerge as consequences of the Riemann-Roch Theorem. The purpose of our work is to apply Flaschka's techniques to operators of order n > 2. We will be defining higher Neumann systems whose theory is closely tied to the spectral theory of linear differential operators of order n. C. Tomei [9], using scattering theory, obtained some of our n = 3 formulas.