Symmetric and smaller PDE for the Ising generating function of cubic maps

Develop a partial differential equation for the Ising generating function I(t,·,·,s) of half-edge-labeled cubic maps that is symmetric in the two variables marking white- and black-monochromatic edges and that is of strictly smaller size (e.g., lower order and/or reduced complexity) than the fourth-order, non-symmetric PDE given in Theorem 3.4.

Background

The paper derives a fourth-order PDE for the generating function I(t,·,·,s) via a change of variables from a KP equation for bipartite maps. This PDE is effective for computation and, combined with mild assumptions, characterizes I. However, it is not symmetric in the two Ising weights, despite I itself being symmetric.

The authors note attempts to construct an alternative PDE that preserves this symmetry and is smaller (in order or complexity), but these attempts have not succeeded, leaving open whether such a PDE exists or can be derived.

References

So far, our efforts to build another PDE that would be both symmetric and smaller have failed.

— The Ising model on cubic maps: arbitrary genus (2504.00768 - Bousquet-Mélou et al., 1 Apr 2025) in Section “A PDE for Ising cubic maps” (immediately after Theorem 3.4)