On Darboux's approach to R-separability of variables I. Isothermic metrics and Dupin-cyclidic metrics

We discuss a problem of the orthogonal R-separability (separability with a factor) of variables in the stationary Schr\"odinger eq. on n-dim. Riemann space following the approach of Gaston Darboux who is the first to formulate a complete theory of the orthogonal R-separability in the Laplace eq. on E^3. According to Darboux the question of R-separability amounts to two conditions: metric has to be isothermic, hence the corresponding parametric surfaces are isothermic in the sense of both classical differential geometry and modern theory of solitons (their curvature nets admit conformal parametrization) and when an isothermic metric is given R-factor is subject to a single constraint which is either functional (R=1) or differential (R-non-trivial). These two conditions are appropriately generalized. Several examples of the approach are discussed in detail. Special attention is paid to the so called Dupin-cyclidic metrics in 3-dim. Riemann space. As a result we re-derive in a systematic way a remarkable result of Darboux: Dupin-cyclidic metrics are (non-regularly) R-separable in the Laplace eq. on E^3. The Appendix is devoted to the early history of Dupin cyclides.