Elliptical galaxies with separable potentials

The perfect ellipsoid is presented: an inhomogeneous triaxial mass model with a gravitational potential that is of Stäckel form. The equations of motion are separable in ellipsoidal coordinates and all stellar orbits have three isolating integrals that are known explicitly. The orbital structure in the perfect ellipsoid is generic for all centrally concentrated triaxial mass models with a finite central density and a Stäckel potential, and it is tractable by analytic means. The orbital shapes are classified in terms of the values of the integrals of motion. Individual orbit densities can be computed without integration of the equations of motion, and some examples are shown. The relevance of mass models with Stäckel potentials for the construction of self-consistent models of elliptical galaxies is discussed.