Lie theory and separation of variables. 9. Orthogonal R-separable coordinate systems for the wave equation ψ_tt-Δ ₂ψ = O

A list of orthogonal coordinate systems which permit R-separation of the wave equation ψ_tt-Δ₂ψ = O is presented. All such coordinate systems whose coordinate curves are cyclides or their degenerate forms are given. In each case the coordinates and separation equations are computed. The two basis operators associated with each coordinate system are also presented as symmetric second order operators in the enveloping algebra of the conformal group O(3,2).