Lattices of Neumann oscillators and Maxwell-Bloch equations

Accepted for publication in Nonlinearity We introduce a family of new non-linear many-body dynamical systems which we call the Neumann lattices. These are lattices of N interacting Neu-mann 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 yields a twisted Poisson structures (in the sense of P. Ševera and A. Weinstein) for the Neu-mann lattices. Thus the Neumann lattices are candidates for integrable systems with twisted Poisson structures.