Lund--Regge Geometry and Integrability of a Generalized Konno--Oono System

We extend recent work on the relation between classical surface theory and partial differential equations, focusing on equations of pseudo-spherical type in the sense of Chern--Tenenblat and on a non-trivial generalization motivated by the Lund--Regge system describing surfaces immersed in $S^3$. As our main application, we study a generalized Konno--Oono system with three dependent variables introduced in a previous paper by one of the authors. We construct an associated parameter-dependent overdetermined linear problem and {\em we establish the existence of infinitely many non-trivial local conservation laws}, hence, integrability. The latter is the most technically demanding part of this paper: it requires a refined analysis of a Riccati pseudo-potential expansion, the use of stereographic coordinates at the full equation manifold level, the construction of special representatives, and a direct proof of non-triviality in horizontal cohomology. We also analyse an illustrative class of travelling wave solutions and show that they can be used to generate surfaces immersed in $S^3$ whose Gaussian curvature changes sign periodically, while their mean curvature are non-vanishing periodic functions. In a limit case, we obtain surfaces that are locally congruent to generalized Clifford tori.