The Navier-Stokes equations for the plane Couette flow are equivariant under
"shift-reflect" and "shift-rotate" transformations:

By “equivariant” we mean that if u(x,t)
is a solution,
so are s_{1}u(x,t),
s_{2}u(x,t),
and s_{1}s_{2}u(x,t);
the four solutions are physically equivalent.
( to see examples of such symmetry-related solutions).
We refer to the space of velocity fields left invariant under the symmetry group
as the S-invariant subspace.
Most of the exact invariant solutions (equilibria,
periodic orbits) shown in what follows belong to this subspace; stable/unstable manifolds, travelling waves do not.