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

Explicit R-matrices for inhomogeneous 3D chiral Potts models: Integrability and the action formulation for