  In this article we consider weak solutions of the three-dimensional
incompressible fluid flow equations with initial data admitting a
one-dimensional symmetry group. We examine both the viscous and inviscid cases.
For the case of viscous flows, we prove that Leray-Hopf weak solutions of the
three-dimensional Navier-Stokes equations preserve initially imposed symmetry
and that such symmetric flows are stable under general three-dimensional
perturbations, globally in time. We work in three different contexts:
two-and-a-half-dimensional, helical and axi-symmetric flows. In the inviscid
case, we observe that, as a consequence of recent work by De Lellis and
Sz\'ekelyhidi, there are genuinely three-dimensional weak solutions of the
Euler equations with two-dimensional initial data. We also present two partial
results where restrictions on the set of initial data, and on the set of
admissible solutions rule out spontaneous symmetry breaking; one is due to
P.-L. Lions and the other is a consequence of our viscous stability result.
