some trial functions in the space of action variables, and $d^{ml}_j(t)$ and $b^{ml}_j(t)$ are time-dependent amplitude functions. On the other hand, one can expand $V_1$ and its corresponding surface density $\Sigma_1$ in the configuration space as $$\begin{aligned}
\Sigma_1(R,\phi,t) &=& \!\!
\sum_{m=-\infty}^{\infty}\sum_{j=0}^{\infty} a^m_j(t)
\sigma^{|m|}_j(R)e^{{\rm i}m\phi},
\label{eq:expansion-sigma1-config} \\
V_1(R,\phi,t) &=& \!\! \sum_{m=-\infty}^{\infty}\sum_{j=0}^{\infty}
a^m_j(t) \psi^{|m|}_j(R)e^{{\rm i}m\phi}. \label{eq:expansion-V1-config}\end{aligned}$$ Here $\psi^{|m|}_j(R)$ and $\sigma^{|m|}_j(R)$ are bi-orthogonal potential–surface density pairs that satisfy the relation $$\label{eq:Djk-definition}
2\pi \int\limits_{0}^{\infty}
\psi^{|m|}_j(R)\sigma^{|m'|}_{j'}(R)R d R = D_j(m)\delta
_{m,m'}\delta_{j,j'},$$ where $\delta_{m,m'}$ is the Kronecker delta and $D_j(m)$ are some constants. It is remarked that the real parts of $\Sigma_1$, $V_1$ and $f_1$ describe physical solutions. In this work I utilize the @CB72 functions $$\begin{aligned}
\psi^{|m|}_j &=&
      -\frac {1}{b} \left (\! {1-\xi \over 2}\! \right )^{1/2} \!\!
             P^{|m|}_i(\xi),~~\xi={R^2-b^2\over R^2+b^2}, \\
\sigma^{|m|}_j &=& \left (\! {2|m|+2j+1\over 2\pi b^2}\! \right )
               \left ( {1-\xi \over 2}\right )^{3/2}\!\!
               P^{|m|}_i(\xi),\end{aligned}$$ that yield [@AI78; @H80] $$D_j(m)=D_j(-m)= -{(2|m|+j)!\over 2b j!}.$$ $P^{|m|}_i(\xi)$ are associated Legendre functions with $i=|m|+j$. Clutton-Brock functions have a length scale $b$, which makes them suitable for reproducing the potential and surface density of soft-centered models. The choice of this parameter is an important step in the calculation of normal modes. I will discuss on this issue later 