Finite dimensional integrable systems related to generalized Schr dinger equations

The binary nonlinearization method is applied to a 4 · ·4 matrix eigenvalue problem. The typical system of the corresponding soliton hierarchy associated with this eigenvalue problem is the multi-component generalization of the nonlinear Schr dinger equation. With this method, Lax pairs and adjoint Lax pairs of the soliton hierarchy are reduced to two classes of finite dimensional Hamiltonian an systems: a spatial finite dimensional Hamiltonian system and a hierarchy of temporal finite dimensional Hamiltonian systems. These finite dimensional Hamiltonian an systems are commutative and Liouville integrable.