The modified tetrahedron equation and its solutions

A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator R-1,R-2,R-3 in the space of a triple Weyl algebra. R-1,R-2,R-3 is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for R-1,R-2,R-3 follows without further calculation. If the Weyl parameter is taken to be a root of unity, R-1,R-2,R-3 decomposes into a matrix conjugation operator R-1,R-2,R-3 and a c-number functional mapping R-1,2,3((f)). The operator R-1,R-2,R-3 satisfies a modified tetrahedron equation (MTE) in which the rapidities are solutions of a classical integrable Hirota-type equations. R-1,R-2,R-3 can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.