Reflection symmetries of Isolated Self-consistent Stellar Systems
  Isolated, steady-state galaxies correspond to equilibrium solutions of the
Poisson--Vlasov system. We show that (i) all galaxies with a distribution
function depending on energy alone must be spherically symmetric and (ii) all
axisymmetric galaxies with a distribution function depending on energy and the
angular momentum component parallel to the symmetry axis must also be
reflection-symmetric about the plane $z=0$. The former result is Lichtenstein's
Theorem, derived here by a method exploiting symmetries of solutions of
elliptic partial differential equations, while the latter result is new. These
results are subsumed into the Symmetry Theorem, which specifies how the
symmetries of the distribution function in configuration or velocity space can
control the planes of reflection symmetries of the ensuing stellar system.
