"Lax Pairs" seem to be the generalization of systems with soliton solutions. Most of the work on Lax Pairs seems to have gone into models of actual physical systems. What I'm looking for is a more methodical mathematical *catalog* of possible Lax Pairs -- whether or not they are physically realizable. I'm curious about the range of behavior that these various soliton solutions can exhibit. I've heard that the *inverse scattering transform* (IST) is a non-linear generalization of the Fourier Transform. I'd also be interested in a *catalog* of IST pairs, analogous to the f(t), FT(f(t)) pairs show in Fourier Transform catalogs. "Toda Lattices" appear to be the finite element versions of the continuous KdV soliton equation. Perhaps Toda Lattices might be capable of Conway "Life" emulation?