  We construct the exact spectral parameter dependent vertex R-matrix for the
classical 3D $\mathcal{N}$-state chiral Potts models, convenient for
considering the model in context of the Bethe ansatz. The R-matrix is defined
on the $\mathcal{N}^4$ dimensional space $V_\mathcal{N}\otimes
V_\mathcal{N}\otimes V_\mathcal{N}\otimes V_\mathcal{N}$, appropriate for
consideration by means of the cube-equations defined in [14]. We present the 2D
quantum spin Hamiltonians for general case and, at $\mathcal{N}=2$, a fermionic
lattice action representation corresponding to 3D Ising's statistical model.
