  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.
