  We present ten new equilibrium solutions to plane Couette flow in small
periodic cells at low Reynolds number (Re) and two new traveling-wave
solutions. The solutions are continued under changes of Re and spanwise period.
We provide a partial classification of the isotropy groups of plane Couette
flow and show which kinds of solutions are allowed by each isotropy group. We
find two complementary visualizations particularly revealing. Suitably chosen
sections of their 3D-physical space velocity fields are helpful in developing
physical intuition about coherent structures observed in low Re turbulence.
Projections of these solutions and their unstable manifolds from their
infinite-dimensional state space onto suitably chosen 2- or 3-dimensional
subspaces reveal their interrelations and the role they play in organizing
turbulence in wall-bounded shear flows.
