How to construct finite-dimensional bi-Hamiltonian systems from soliton equations: Jacobi integrable potentials

A systematic method of constructing finite-dimensional integrable systems starting from a bi-Hamiltonian hierarchy of soliton equations is introduced. The existence of two Hamiltonian structures of the hierarchy leads to a bi-Hamiltonian formulation of the resulting finite-dimensional systems. The case of coupled KdV hierarchies is studied in detail. A surprising connection with separable Jacobi potentials is uncovered and described.