Meridional velocity isotropy of a two-integral distribution ($⟨v_R^2⟩=⟨v_z^2⟩$)

At fixed position a two-integral distribution depends on $v_R,v_z$ only through $v_R^2+v_z^2$. Swapping these integration variables makes their second moments equal. Their odd moments and mixed second moments vanish by reflection symmetry. For an even function of $L_z$ the azimuthal velocity is also centered, so all mixed entries of the velocity ellipsoid vanish.