  The purpose of this work is to review the status about stationary solutions
of the axially symmetric Einstein-Vlasov system with a focus on open problems
of both analytical and numerical nature. For the latter we emphasize that the
code used to construct stationary solutions in \cite{Ames2016,Ames2019} is open
source, see \cite{Ames2023joss}. In the analytical setting the open problems
include establishing methods for proving existence of axisymmetric stationary
solutions which are far from spherically symmetric, both in the general case
and for certain special classes of solutions pointed out in the text. In the
numerical setting there are intriguing properties of highly relativistic
solutions that demand further attention, such as whether a sequence of such
stationary solutions can approach a Kerr black hole, or if they necessarily
approach the thin ring limit reminiscent of cosmic strings. The question of
whether stationary solutions include states with thin-disk like morphologies as
seen in many galaxies is also open. Finally, there are opportunities to extend
this research to new settings such as the case of massless particles and
coupled black hole-matter systems. We believe that some of the open problems
highlighted here are of central importance for the understanding of nature.
