On the lattice-geometry and birational group of the six-point multi-ratio equation

The inherent self-consistency properties of the six-point multi-ratio equation allow it to be considered on a domain associated with a T-shaped Coxeter–Dynkin diagram. This extends the Kadomtsev–Petviashvili lattice, which has AN symmetry, and incorporates also Korteweg–de Vries-type dynamics on a sub-domain with DN symmetry, and Painlevé dynamics on a sub-domain with E∼8 symmetry. More generally, it can be seen as a distinguished representation of Coble’s Cremona group associated with invariants of point sets in projective space.