Bäcklund transformations and three-dimensional lattice equations

A (nonlocal) linear integral equation is studied, which allows for Bäcklund transformations in the measure. The compatibility of three of these transformations leads to an integrable nonlinear three-dimensional lattice equation. In appropriate continuum limits the two-dimensional Toda-lattice equation and the Kadomtsev-Petviashvili equation are derived as special examples