Integrability in $[d+1]$ dimensions: combined local equations and commutativity of the transfer matrices

We propose new inhomogeneous local integrability equations - combined equations, for statistical vertex models of general dimensions in the framework of the Algebraic Bethe Ansatz (ABA). For the low dimensional cases the efficiency of the step by step consideration of the transfer matrices’ commutation is demonstrated. We construct some simple 3D solutions with the three-state $R$-matrices of certain 20-vertex structure; the connection with the quantum three-qubit gates is discussed. New, restricted versions of 3D local integrability equations with four-state $R$-matrices are defined, too. Then we construct a new 3D analogue of the two-dimensional star-triangle equations.