Geometry of non-regular separation

A geometrical setting for the notion of non-regular additive separation for a PDE, introduced by Kalnins and Miller, is given. This general picture contains as special cases both fixed-energy separation and constrained separation of Helmholtz and Schodinger equations (not necessarily orthogonal). The geometrical approach to non-regular separation allows to explain why it is possible to find some coordinates in Euclidean 3-space where the R-separation of Helmholtz equation occurs, but it depends on a lower number of parameters than in the regular case, and it is apparently not related to the classical Stackel form of the metric.