Maximally prograde stellar distribution ($F_{+}(E,L_z)=2F(E,L_z)\Theta (L_z)$)

For a nonrotating axisymmetric galactic distribution function even in $L_z$, reverse every retrograde azimuthal velocity. The resulting distribution is $F_{+}(E,L_z)=2F(E,L_z)\Theta (L_z)$: it preserves density and even velocity moments but makes the streaming velocity $⟨|v_{\varphi }|⟩$. Dependence on conserved $E,L_z$ preserves stationarity. A centered Gaussian distribution of width $\sigma$ becomes a half-normal distribution, with mean $\sqrt{2/\pi }\sigma$.