  Recent progress indicates that highly symmetric recurring solutions of the
Navier-Stokes equations, such as equilibria and periodic orbits, provide a
skeleton for turbulence dynamics in state-space. Many of these solutions have
been found for flat-walled plane Couette, channel, and pipe flows. Rough-walled
flows are of great practical significance, yet no recurring solutions are known
for these flows. We present a numerical homotopy method to continue solutions
from flat-walled plane Couette flow (PCF) to grooved PCF, demonstrated here at
a Reynolds number of 400, to act as a starting point for similar continuation
to rough-walled flows. Loss of spanwise homogeneity in grooved PCF reduces
continuous families of solutions (identical up to translational shifts) in
flat-walled Couette flow to multiple, discrete families in grooved Couette
flow; this can manifest in turbulence as spatially anchored exact coherent
structures near the wall, so that turbulent statistics reflect
symmetry-restricted structure of exact recurring solutions. Furthermore, the
vortex-streak structures characteristic of these equilibria are squeezed out of
the grooves when the groove-wavelength is smaller than the characteristic
spanwise size of the structures. This produces reduced shear stress at the wall
even at the low Reynolds numbers considered, and the mechanism is consistent
with the drag reduction observed in some riblet-mounted turbulent flows.
