= Velocity-ellipsoid alignment in a Stäckel potential
{title2=$\langle v_i v_j\rangle=0\quad(i\ne j)$}
A regular phase-mixed distribution $F(H,p_\phi,I)$ in an axisymmetric <Stäckel potential> is even separately in $v_\lambda$ and $v_\mu$, because these <integrals> depend on those components only through squares. Odd <integration> then gives all three mixed <second moments> zero, even if the $p_\phi$ dependence produces azimuthal streaming. Bare stationarity alone is insufficient in degenerate resonant potentials with additional <integrals>. For example the isotropic harmonic <Hamiltonian> has the extra conserved $K=v_xv_y+\omega^2xy$; a positive Gaussian $\exp(-aH+\epsilon K)$ with $a>|\epsilon|>0$ is a steady tracer and has nonzero $\langle v_xv_y\rangle$. Its harmonic <gravitational potential> is a <Stäckel potential> in these spheroidal coordinates. This tracer need not itself source that harmonic <gravitational potential>. The alignment claim thus needs the phase-mixed three-integral hypothesis, not merely the weak form of <Jeans theorem>.
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