Discrete versions of some classical integrable systems and factorization of matrix polynomials (Q1180975)

This remarkable paper studies discrete versions of classical integrable systems: the free rigid body, the Heisenberg chain with classical spins, and a system on Stiefel manifolds. The integrability follows from a Lax representation which in turn is found from factorizations of matrix polynomials. This paper gives a method to discretize integrable equations and at the same time to preserve the integrability under discretization. This is a difficult problem since obvious guesses don't usually work. The authors open herewith a new field of investigation in the theory of integrable systems. The dynamics is completely described by Abelian functions and the flow is linear on a Prym variety associated to the spectral curve of the problem. The paper ends with an interesting application of the method to the billiard problem in the interior of an \(N\)-dimensional ellipsoid.