On the continuous limit of integrable lattices II. Volterra systems and sp(N) theories

A connection is suggested between the zero-spacing limit of a generalized N-fields Volterra (V_N) lattice and the KdV-type theory which is associated, in the Drinfeld-Sokolov classification, to the simple Lie algebra sp(N). The recombination method developed in a previous paper of ours is applied to study in full detail the case N=2: the infinitely many commuting vector fields, the Hamiltonian structure and the Lax formulation of the corresponding Volterra system V_2 are shown to give in the continuous limit the homologous sp(2) KdV objects, through specified operations of field rescaling and recombination. Finally, the case of arbitrary N is attacked, showing how to obtain the sp(N) Lax operator from the continuous limit of the V_N system.