Integrable theories allow to solve nonlinear classical dynamics in terms of scattering data for a linear system. In TGD framework this translates to quantum classical correspondence. The solutions of modified Dirac equation define the scattering data. The conjecture is that octonionic real-analyticity with space-time surfaces identified as surfaces for which the imaginary part of the biquaternion representing the octonion vanishes solves the field equations. This conjecture generalizes the conformal invariance to its octonionic analog. If this conjecture is correct, the scattering data should define a real analytic function whose octonionic extension defines the space-time surface as a surface for which its imaginary part in the representation as bi-quaternion vanishes. There are excellent hopes about this thanks to the reduction of the modified Dirac equation to geometric optics.
I do not bother to type 10 pages of text here but refer to the article An attempt to understand preferred extremals of Kähler action and to the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts of "Physics as Generalized Number Theory".