We show that , provided the principal axes of the second velocity moment tensor of a stellar population are generally unequal and are oriented perpendicular to a set of orthogonal surfaces at each point , then those surfaces must be confocal quadric surfaces and the potential must be separable or Stäckel .
This is true under the mild assumption that the even part of the distribution function ( DF ) is invariant under time reversal v _ { i } \rightarrow - v _ { i } of each velocity component .
In particular , if the second velocity moment tensor is everywhere exactly aligned in spherical polar coordinates , then the potential must be of separable or Stäckel form ( excepting degenerate cases where two or more of the semiaxes of ellipsoid are everywhere the same ) .
The theorem also has restrictive consequences for alignment in cylindrical polar coordinates , which is used in the popular Jeans Anisotropic Models ( JAM ) of Cappellari ( 2008 ) .

We analyse data on the radial velocities and proper motions of a sample of \sim 7400 stars in the stellar halo of the Milky Way .
We provide the distributions of the tilt angles or misalignments from the spherical polar coordinate systems .
We show that in this sample the misalignment is always small ( usually within 3 ^ { \circ } ) for Galactocentric radii between \sim 7 and \sim 12 kpc .
The velocity anisotropy is very radially biased ( \beta \approx 0.7 ) , and almost invariant across the volume in our study .
Finally , we construct a triaxial stellar halo in a triaxial NFW dark matter halo using a made-to-measure method .
Despite the triaxiality of the potential , the velocity ellipsoid of the stellar halo is nearly spherically aligned within \sim 6 \mathrm { degrees } for large regions of space , particularly outside the scale radius of the stellar halo .
We conclude that the second velocity moment ellipsoid can be close to spherically aligned for a much wider class of potentials than the strong constraints that arise from exact alignment might suggest .