Non-regular and non-Stäckel R-separation for 3-dimensional Helmholtz equation and cyclidic solitons of wave equation

We discuss a class of orthogonal coordinates in E3 which are non-trivially R-separable for 3-dimensional Euclidean Helmholtz equation. The separation is both non-regular (in the sense of Kalnins and Miller) and non-Stäckel. One family of parametric surfaces consists of parallel Dupin cyclides, the other two consist of circular cones. These coordinates are used to simplify derivation of Friedlander's formulae for modulated progressive wave solutions to the wave equation and to correct some errors of his paper.