Nonorthogonal R�separable coordinates for four�dimensional complex Riemannian spaces

We classify all Râ€�separable coordinate systems for the equations Δ₄Ψ=J⁴ _{i, j=1}g^−1/2 ∂_ j(g^1/2g^i j∂_iΨ) =0 and J⁴ _{i, j=1}g^i j∂_iW∂_ jW =0 with special emphasis on nonorthogonal coordinates, and give a group theoretic interpretation of the results. For flat space we show that the two equations separate in exactly the same coordinate systems and present a detailed list of the possibilities. We demonstrate that every Râ€�separable system for the Laplace equation Δ₄Ψ=0 on a conformally flat space corresponds to a separable system for the Helmholtz equations Δ₄Φ=λΦ on one of the manifolds E₄, S₁×S₃, S₂×S₂, and S₄.