News on integrable systems: Discrete Heisenberg Chain for NXNmatrices, BTs for BC and trigonometric Gaudin models, Bertrand systems incurved space.

Abstract:

Some recent results on integrable models are briefly reviewed. We start with discrete systems, exhibiting (i) a differential difference version of the Heisenberg ferromagnet for NXN cyclic matrices, derived in the framework of the theory of Poisson-Nijenhuis manifolds (with F.Zullo) and (ii) a scheme for constructing BTs for the 2X2 trigonometric Gaudin. Then we illustrate the main results contained in a paper jointly written with A.Ballesteros,F.Herranz and A.Enciso where the analogues of the Kepler and harmonic oscillator systems in appropriate curved manifolds are derived, and their superintegrability is discussed.