The stationary Navier-Stokes equations under Navier boundary conditions are considered
in a square. The uniqueness of solutions is studied in dependence of the Reynolds number and
of the strength of the external force. For some particular forcing, it is shown that uniqueness
persists on some continuous branch of solutions, when these quantities become arbitrarily
large. On the other hand, for a different forcing, a branch of symmetric solutions is shown
to bifurcate, giving rise to a secondary branch of nonsymmetric solutions. This proof is
computer-assisted, based on a local representation of branches as analytic arcs.