During years evidence supporting the idea that TGD could be an integrable theory in some sense has accumulated. The challenge is to show that various ideas about what integrability means form pieces of a bigger coherent picture. Of course, some of the ideas are doomed to be only partially correct or simply wrong. Since it is not possible to know beforehand what ideas are wrong and what are right the situation is very much like in experimental physics and it is easy to claim (and has been and will be claimed) that all this argumentation is useless speculation. This is the price that must be paid for the luxury of genuine thinking.
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.
For details and background the reader can consult 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.