  Flexibility and rigidity properties of steady (time-independent) solutions of
the Euler, Boussinesq and Magnetohydrostatic equations are investigated.
Specifically, certain Liouville-type theorems are established which show that
suitable steady solutions with no stagnation points occupying a two-dimensional
periodic channel, or axisymmetric solutions in (hollowed out) cylinder, must
have certain structural symmetries. It is additionally shown that such
solutions can be deformed to occupy domains which are themselves small
perturbations of the base domain. As application of the general scheme, Arnol'd
stable solutions are shown to be structurally stable.
