Abstract For two-dimensional lattice equations the standard definition of integrability is that it should be possible to extend the map consistently to three dimensions, i.e., that it is “consistent around a cube” (CAC). Recently Adler, Bobenko and Suris conducted a search based on this principle, together with the additional assumptions of symmetry and “the tetrahedron property”. We present here results of a search for CAC lattice equations assuming also the same symmetry properties, but not… Expand