Lattices of Neumann oscillators and Maxwell-Bloch equations

We introduce a family of new nonlinear many-body dynamical systems which we call the Neumann lattices. These are lattices of N interacting Neumann oscillators. The interactions are of magnetic type. We construct large families of conserved quantities for the Neumann lattices. For this purpose we develop a new method of constructing the first integrals which we call the reduced curvature condition. Certain Neumann lattices are natural partial discretizations of the Maxwell-Bloch equations. The Maxwell-Bloch equations have a natural Hamiltonian structure whose discretizations yield twisted Poisson structures (as described by. Severa and Weinstein) for the Neumann lattices. Thus the Neumann lattices are candidates for integrable systems with twisted Poisson structures.