Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality

It is shown that the discrete Darboux system, descriptive of conjugate latt ices in Euclidean spaces, admits constraints on the (adjoint) eigenfunction s which may be interpreted as discrete orthogonality conditions on the latt ices. Thus, it turns out that the elementary quadrilaterals of orthogonal l attices are cyclic. Orthogonal lattices on lines, planes and spheres are di scussed and the underlying integrable systems in one, two and three dimensi ons are derived explicitly. A discrete analogue of Bianchi's Ribaucour tran sformation is set down and particular orthogonal lattices are given. As a b y-product, discrete Dini surfaces are obtained.