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 (DF) depending on energy alone f(E) must be spherically symmetric, and (II) all axisymmetric galaxies with a DF depending on energy and the angular momentum component parallel to the symmetry axis f(E, L_z) must also be reflection-symmetric about the plane z = 0. The former result is known, whilst the latter result is new. These results are subsumed into the Symmetry Theorem, which specifies how the symmetries of the DF in configuration or velocity space can control the planes of reflection symmetries of the ensuing stellar system.