A second look at the Kurth solution in galactic dynamics

The Kurth solution is a particular non-isotropic steady state solution to the gravitational Vlasov-Poisson system. It has the property that by means of a suitable time-dependent transformation it can be turned into a family of time-dependent solutions. Therefore, for a general steady state \begin{document}$ Q(x, v) = \tilde{Q}(e_Q, \beta) $\end{document} , depending upon the particle energy \begin{document}$ e_Q $\end{document} and \begin{document}$ \beta = \ell^2 = |x\wedge v|^2 $\end{document} , the question arises if solutions \begin{document}$ f $\end{document} could be generated that are of the form\begin{document}$ f(t) = \tilde{Q}\Big(e_Q(R(t), P(t), B(t)), B(t)\Big) $\end{document}for suitable functions \begin{document}$ R $\end{document} , \begin{document}$ P $\end{document} and \begin{document}$ B $\end{document} , all depending on \begin{document}$ (t, r, p_r, \beta) $\end{document} for \begin{document}$ r = |x| $\end{document} and \begin{document}$ p_r = \frac{x\cdot v}{|x|} $\end{document} . We are going to show that, under some mild assumptions, basically if \begin{document}$ R $\end{document} and \begin{document}$ P $\end{document} are independent of \begin{document}$ \beta $\end{document} , and if \begin{document}$ B = \beta $\end{document} is constant, then \begin{document}$ Q $\end{document} already has to be the Kurth solution. This paper is dedicated to the memory of Professor Robert Glassey.