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

We introduce a systematic method of constructing finite-dimensional integrable systems starting from a bi-hamiltonian hierarchy of soliton equations. The existence of two hamiltonian structures of the hierarchy lead 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. (authors)