It was in 1973 that the fundamental AKNS spectral system was set down. This linear representation for a wide class of soliton equations yields up via compatibility conditions, in particular, the canonical sine-Gordon, mKdV, KdV and NLS equations. Hard upon this work, in 1976, came that of Lund and Regge and Pohlmeyer, which established a connection between the geometry of privileged classes of surfaces and soliton theory. Thus, what is now known as the Pohlmeyer-Lund-Regge solitonic system was generated via a Gauss-Mainardi-Codazzi system, with the corresponding Gauss-Weingarten equations viewed as a 3 × 3 linear representation. Moreover, Lund and Regge adopted a spinor formulation to make direct connection with 2 × 2 representations as embodied in the AKNS system.

The next major development in the geometry of soliton theory came in 1982 with the pioneering work of Sym when he introduced the notion of soliton surfaces. Thus, the soliton surfaces associated with the sine-Gordon and NLS equations are, in turn, the pseudospherical and Hasimoto surfaces. In general, the one-parameter class of soliton surfaces: Σ: r = r (u, v) associated with a particular solution of a solitonic equation is obtained by insertion of that solution into the relevant Gauss-Weingarten equations and integration thereof to determine the position vector r. However, this direct integration approach may be circumvented by an ingenious method described in. This Sym-Tafel procedure depends crucially on the presence of a ‘spectral parameter’ in the linear representation for the soliton equation in question.