  A specific family of spanwise-localised invariant solutions of plane Couette
flow exhibits homoclinic snaking, a process by which spatially localised
invariant solutions of a nonlinear partial differential equation smoothly grow
additional structure at their fronts while undergoing a sequence of saddle-node
bifurcations. Homoclinic snaking is well understood in the context of simpler
pattern forming systems such as the one-dimensional Swift-Hohenberg equation
with cubic-quintic nonlinearity, whose solutions remarkably well resemble the
snaking solutions of plane Couette flow. We study the structural stability of
the characteristic snakes-and-ladders structure associated with homoclinic
snaking for flow modifications that break symmetries of plane Couette flow. We
demonstrate that wall-normal suction modifies the bifurcation structure of
three-dimensional plane Couette solutions in the same way, a symmetry-breaking
quadratic term modifies solutions of the one-dimensional Swift-Hohenberg
equation. These modifications are related to the breaking of the discrete
rotational symmetry. At large amplitudes of the symmetry-breaking wall-normal
suction the connected snakes-and-ladders structure is destroyed. Previously
unknown solution branches are created and can be parametrically continued to
vanishing suction. This yields new localised solutions of plane Couette flow
that exist in a wide range of Reynolds number.
