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_1,2,3 in the space of a triple Weyl algebra. R_1,2,3 is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for R_1,2,3 follows without further calculation. If the Weyl parameter is taken to be a root of unity, R_1,2,3 decomposes into a matrix conjugation operator R _1,2,3 and a c-number functional mapping R_1,2,3 ^(f). The operator R_1,2,3 satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equations. R_1,2,3 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.