Symmetries, separability and volume forms

In order to clarify certain misconceptions in the literature, we discuss th e details of the way determining equations for general symmetries of a Lagr angian system differ from determining equations for Noether symmetries, and establish the minimal set of extra conditions which have to be imposed in practical situations for the former to reduce to the latter. We further der ive properties by which, in situations where the system is integrable throu gh quadratic integrals in involution, the components of the corresponding N oether symmetries themselves can be used to compute the separation variable s, if they exist, for the Hamilton-Jacobi equation. (C) 1998 American Insti tute of Physics. [S0022-2488(98)02711-X].