Tetrahedron maps and symmetries of three dimensional integrable discrete equations

A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on Z(3) is investigated. Our approach is a generalization of a method developed in the context of Yang-Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case-by-case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations. Published under license by AIP Publishing.