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The inherent self-consistency properties of the six-point multi-ratioequation allow it to be considered on a domain associated with a T-shapedCoxeter-Dynkin diagram. This extends the KP lattice, which has A_N symmetry,and incorporates also KdV-type dynamics on a sub-domain with D_N symmetry, andPainleve dynamics on a sub-domain with affine-E8 symmetry. More generally, itcan be seen as a distinguished representation of Coble's Cremona groupassociated with invariants of point sets in projective space.

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