Hamiltonian structures for integrable field theory models. II. Models with $O(n)$ and $Sp(2k)$ symmetry on a one-dimensional lattice

A new family of classical integrable systems with $O(n)$ and $Sp(2k)$ symmetry is found. It is shown that these systems can be regarded as lattice analogs of models of the nonlinear Schrödinger equation on symmetric spaces. An example of a $O(n)$-invariant classical discrete magnet with local Hamiltonian is constructed.