Classification of 3D Consistent Quad-Equations

Abstract

We consider 3D consistent systems of six possibly different quad-equations assigned to the faces of a cube. The well-known classification of 3D consistent quad-equations, the so-called ABS-list, is included in this situation. The extension of these equations to the whole lattice Z 3 is possible by reflecting the cubes. For every quad-equation we will give at least one system included leading to a Bcklund transformation and a zero-curvature representation which means that they are integrable.