Relation of (semi)-discrete B-series Toda systems to the discrete Moutard equation and discrete Königs nets

In the continuous theory, symmetry reductions of the open Toda lattice yield exponential systems associated with Cartan matrices of B- and C-type; these connect to classical equations such as the Goursat and Moutard equations, and geometrically to Königs nets. The paper establishes discrete Darboux formulae for linear hyperbolic operators with finite Laplace series, paralleling the continuous theory.

For the discrete theory, C-series reductions on the A-series lattice are known, but the status of B-series reductions and their geometric/analytic links remain unsettled. Specifically, the authors highlight that it is currently not known whether the (semi)-discrete B-series Toda systems constructed in the integrable-systems literature are related to the discrete Moutard equation and to discrete Königs nets, leaving a conceptual gap between discrete integrable lattices and discrete differential geometry.