Orbit volume in an axisymmetric Stäckel potential ($p_{\tau }^2\ge 0$)

For fixed $E,l,I$, separated motion obeys $p_{\tau }^2=[E\tau +G(\tau )-I+(\alpha -\beta )l^2/(2(\tau +\alpha ))]/[2(\tau +\alpha )(\tau +\beta )]$. Allowed intervals are where this is nonnegative, with turning points and coordinate-fold endpoints treated separately. A generic nonresonant bound orbit with $l\ne 0$ circulates azimuthally and densely samples a projected volume bounded by confocal spheroids and hyperboloids. It is not uniformly distributed in volume. Resonant or zero-angular-momentum orbits can project to lower-dimensional regions; integrability alone does not imply that every orbit fills a three-dimensional shell.