It is straightforward to generalize the proof to the instance of axisymmetric stellar systems with second velocity moments aligned in spheroidal coordinates (λ, μ, ϕ). This coordinate system has been introduced in equation (13) and is described in detail in, for example, Morse & Feshbach (1953) or Binney & Tremaine (2008). Here, we will show that if the second velocity moment ellipsoid is everywhere aligned in spheroidal coordinates (〈vλvϕ〉 = 〈vλvμ〉 = 〈vμvϕ〉 = 0), then the gravitational potential has Stäckel or separable form. Again, we construct the even part of the DF. Introducing canonical coordinates, the DF has the form ${F_{\rm e}}(p_\lambda ^2, p_\mu ^2, p_\phi ^2, \lambda ,\mu ,\phi )$. The Hamiltonian is

(44)

\begin{equation}
H = {1\over 2} \Bigl ( {p_\lambda ^2\over P^2} + {p_\mu ^2 \over Q^2} + {p_\phi ^2 \over R^2} \Bigr ) - \psi (\lambda ,\mu ,\phi ),
\end{equation}

where the scale factors are P, Q and R are

(45)

\begin{eqnarray}
P^2 &amp;=&amp; {\lambda -\mu \over 4(\lambda + a)(\lambda +b)},\qquad Q^2 = {\mu -\lambda \over 4(\mu + a)(\mu +b)},\nonumber \\
R^2 &amp;=&amp; {(\lambda +a)(\mu +a)\over a-b},
\end{eqnarray}

and a and b are constants (see for example the tables of Lynden-Bell 1962 or equation (6) of Evans & Lynden-Bell 1989 or Section 2 of de Zeeuw 1985). This implies that the DF can be re-written as ${F_{\rm e}}(H, p_\mu ^2, p_\phi ^2, \lambda ,\mu ,\phi ).$ Just as before, requiring the Poisson bracket {H, Fe} to vanish implies that ∂Fe/∂λ also vanishes and so Fe is independent of λ. This leaves us with the condition

(46)

\begin{equation}
A {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} p_\mu } - {\mathrm{\partial} \psi \over \mathrm{\partial} \phi } {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} p_\phi } = {p_\mu \over Q^2} {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} \mu } + {p_\phi \over R^2} {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} \phi }
\end{equation}

with

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\begin{eqnarray}
&amp;&amp;{A ={1\over \lambda -\mu }}\nonumber\\
&amp;&amp;{\left[ {p_\mu ^2\over 2} {\mathrm{\partial} \over \mathrm{\partial} \mu }\left( {\lambda -\mu \over Q^2} \right) \!+\! {p_\phi ^2\over 2} {\mathrm{\partial} \over \mathrm{\partial} \mu } \left( {\lambda \!-\!\mu \over R^2} \right) - {\mathrm{\partial} \over \mathrm{\partial} \mu }\left( (\lambda -\mu ) \psi \right) +H\right].}\nonumber\\
\end{eqnarray}

Again, the equation must hold on transforming pμ → −pμ, so that in the general case (i.e. ignoring degenerate cases like isotropy), we must have

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\begin{equation}
A {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} p_\mu } = {p_\mu \over Q^2} {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} \mu }, \qquad \qquad {\mathrm{\partial} \psi \over \mathrm{\partial} \phi } {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} p_\phi }= {p_\phi \over R^2} {\mathrm{\partial} {F_{\rm e}}\over \mathrm{\partial} \phi }.
\end{equation}

Multiplying the first equation by λ − μ, differentiating with respect to λ at constant H and then using the definitions of the scale factors gives us the simple result

(49)

\begin{equation}
{\mathrm{\partial} ^2 \over \mathrm{\partial} \lambda \mathrm{\partial} \mu } \left( (\lambda -\mu )\psi \right) =0.
\end{equation}

Integrating up, this gives us

(50)

\begin{equation}
\psi = {A(\lambda ,\phi ) - B (\mu ,\phi ) \over \lambda - \mu },
\end{equation}

where A(λ, ϕ) and B(μ, ϕ) are arbitrary.
Citation: de Zeeuw 1985
Citation function: Uses