the limiting cases of rosette orbits with $J_\phi=0$ and $J_R=0$, respectively. By representing $f$ in terms of the action-angle variables, the CBE reads $$\frac{\partial f}{\partial t}+[ f,{\cal H} ]=0,
\label{eq:CBE-action-angle}$$ where $[,]$ denotes a Poisson bracket taken over the action-angle space. According to Jeans theorem [@J1915; @L-B62] $f_0$ depends on the phase space coordinates through the integrals of motion, which are the actions in the present formulation, and one obtains $[f_0,{\cal H}_0]=0$. Subsequently, equation (\[eq:CBE-action-angle\]) may be rewritten as $$\frac{\partial f_1}{\partial t}=-\left [f_1,{\cal H}_0 \right ]-
\left [ f_0,{\cal H}_1 \right ]-\left [f_1,{\cal H}_1 \right ],
\label{eq:CBE-perturbed}$$ where ${\cal H}_1$ is the perturbed Hamiltonian. A dark matter halo contributes both to ${\cal H}_0$ and to ${\cal H}_1$ if it is live, i.e., if it exchanges momentum/energy with the luminous stellar component. In this paper I confine myself to a rigid halo that only contributes to ${\cal H}_0$ through $V_0$ and assume that ${\cal H}_1=V_1$ is the perturbed potential due to self-gravity.

Let me expand $f_1$ and $V_1$ in Fourier series of angle variables and write $$\begin{aligned}
f_1(\Theta,\textbf{\textit{J}},t) &=& \!\!
\sum_{m,l=-\infty}^{\infty}\sum_{j=0}^{\infty} d^{ml}_j(t)
\Phi^{ml}_j(\textbf{\textit{J}})e^{{\rm i}\left(m\theta_\phi+l\theta_R
\right)},
\label{eq:expansion-f1} \\
V_1(\Theta,\textbf{\textit{J}},t) &=& \!\!
\sum_{m,l=-\infty}^{\infty}\sum_{j=0}^{\infty} b^{ml}_j(t)
\Psi^{ml}_j(\textbf{\textit{J}})e^{{\rm i}\left(m\theta_\phi+l\theta_R
\right)}, \label{eq:expansion-V1}\end{aligned}$$ where $\Phi^{ml}_j(\textbf{\textit{J}})$ and $\Psi^{ml}_j(\textbf{\textit{J}})$ are 