  Spatially extended systems, such as channel or pipe flows, are often
equivariant under continuous symmetry transformations, with each state of the
flow having an infinite number of equivalent solutions obtained from it by a
translation or a rotation. This multitude of equivalent solutions tends to
obscure the dynamics of turbulence. Here we describe the `first Fourier mode
slice', a very simple, easy to implement reduction of SO(2) symmetry. While the
method exhibits rapid variations in phase velocity whenever the magnitude of
the first Fourier mode is nearly vanishing, these near singularities can be
regularized by a time-scaling transformation. We show that after application of
the method, hitherto unseen global structures, for example Kuramoto-Sivashinsky
relative periodic orbits and unstable manifolds of travelling waves, are
uncovered.
