PARTIAL-DERIVATIVE-REDUCTIONS OF THE MULTIDIMENSIONAL QUADRILATERAL LATTICE - THE MULTIDIMENSIONAL CIRCULAR LATTICE

We apply a recently introduced [21, 15] reduction method, based on the partial derivative-dressing, to construct a large class of integrable reductions of the equations characterizing the multidimensional quadr ilateral lattice tan N-dimensional lattice in R-M, N less than or equa l to M, whose elementary quadrilaterals are planar and whose continuou s limit describes submanifolds parametrized by conjugate lines [11]). We also show that, generically, in the limit of the small lattice para meter, half of these reductions lead to the Darboux equations for symm etric fields and the second half lead to the generalized Lame equation s describing N-dimensional submanifolds of E-M parametrized by conjuga te orthogonal systems of coordinates. We finally show that a distingui shed example of the second class of reductions corresponds to the mult idimensional circular lattice tan N-dimensional lattice in E-M, N less than or equal to M, whose elementary quadrilaterals are inscribed in circles [7]).