Neumann and Bargmann Systems Associated with an Extension of the Coupled KdV Hierarchy

An eigenvalue problem with a reference function and the corresponding hierarchy of nonlinear evolution equations are proposed. The bi-Hamiltonian structure of the hierarchy is established by using the trace identity. The isospectral problem is non-linearized as to be finite-dimensional completely integrable systems in Liouville sense under Neumann and Bargmann constraints.