  The evolution of a rotating axisymmetric galaxy from an asymmetric state to a
state of mirror symmetry with respect to the galactic plane has as basic result
that in the asymmetric initial state the perpendicular $z$ normal mode is
unstable for the $1:1$ and $1:2$ resonances. Dynamically this results in a
transfer of mass and momenta towards the galactic plane. The timescale of
evolution to symmetric equilibrium will determine in these cases the final
distribution function describing position and velocities. In the case of the
$1:1$ resonance we have in the final stage apart from the angular momentum
integral 2 adiabatic invariants describing nonlinear dynamics. For the $1:2$
resonance the dynamics in the final stage is simpler, apart from the angular
momentum integral the dynamics is governed by the 2 actions. In the first
sections the results of mathematical analysis have been summarised, examples of
evolution are given in section 3.
