Principal velocity surfaces in stellar dynamics

Recent work by An and Evans has revived interest in the coordinate surfaces that lie along the principal axes of the stress tensor. Here we complete our list of non-axially symmetric systems with local integrals and prove that those principal velocity surfaces exist for all systems with three local integrals. We then demonstrate how systems in non-separable potentials evade the Eddington–An theorem by having distribution functions that lack perfect reflection symmetry in a meridional velocity component, and discuss other consequences of this remarkable theorem.

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