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 $Q(x, v)=\tilde{Q}(e_Q, \beta)$, depending upon the particle energy $e_Q$ and $\beta=\ell^2=|x\wedge v|^2$, the question arises if solutions $f$ could be generated that are of the form $$f(t)=\tilde{Q}\Big(e_Q(R(t), P(t), B(t)), B(t)\Big)$$ for suitable functions $R$, $P$ and $B$, all depending on $(t, r, p_r, \beta)$ for $r=|x|$ and $p_r=\frac{x\cdot v}{|x|}$. We are going to show that, under some mild assumptions, basically if $R$ and $P$ are independent of $\beta$, and if $B=\beta$ is constant, then $Q$ already has to be the Kurth solution. This paper is dedicated to the memory of Professor Robert Glassey.