Neumann Systems for the Algebraic AKNS Problem

The Neumann system, an algebraically completely integrable Hamiltonian system, consists of harmonic oscillators constrained to move on the unit spere in configuration space. Any finite gap potential of Hill's equation may be expressed in terms of a solution of the Neumann problem. The present work is concerned with an algebraically completely integrable Hamiltonian system whose solutions may be used to describe the finite gap solutions of the AKNS spectral problem, a first order two-by-two matrix linear system. Trace formulas, constraints, Lax paris, and constants of motion are obtained using Krichever's algebraic inverse spectral transform. Computations are carried out explicityly over the class of spectral problems with square matrix coefficients.