Surfaces of Demoulin: differential geometry, Bäcklund transformation and integrability

The surfaces of Demoulin constitute an important subclass of surfaces in projective differential geometry which arise in many seemingly unrelated geometric constructions. Analytically, they are described by a two-component system which coincides with the D₃⁽¹⁾ Toda lattice. We review some of the most important geometric properties of the Demoulin surfaces and construct a Bäcklund transformation which may be specialized to the well-known Bäcklund transformation for the Tzitzeica equation governing affine spheres in affine geometry.