On the continuous limit of integrable lattices III. Kupershmidt systems and sl(N+1) KdV theories

We discuss the connection between the zero-spacing limit of the N- fields Kupershmidt lattice and the KdV-type theory corresponding to the Lie algebra sl(N + 1). The case N = 2 is worked out in detail, recovering from the Limit process the Boussinesq theory with its infinitely many commuting vector fields, their Lax pairs and Hamiltonian formulations. The 'recombination method' proposed here to derive the Boussinesq hierarchy from the limit of the N = 2 Kupershmidt system works, in principle, for arbitrary N.