THE INVERSE SCATTERING METHOD AND SYSTEMS OF EQUATIONS QXY = F(Q,QY)

Consider equations of the form q(xy) = F(q, q(y)), () where q is in g eneral a complex vector and the function F depends nontrivially both o n q and on q(y). We show that a family S of such equations can be inve stigated by the inverse scattering method. If an equation () belongs to S, the function F depends linearly on q and algebraically on q(y). We show that the family S contains a subfamily in which each equation can be obtained from the two dimensional Toda lattice equations by a B acklund transformation.