Tetrahedron maps and symmetries of three dimensional integrable discrete equations

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A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on Z3 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.