  We consider the planar Taylor-Couette system for the steady motion of a
viscous incompressible fluid in the region between two concentric disks, the
inner one being at rest and the outer one rotating with constant angular speed.
We study the uniqueness and multiplicity of solutions to the forced system in
different classes. For any angular velocity we prove that the classical
Taylor-Couette flow is the unique smooth solution displaying rotational
symmetry. Instead, we show that infinitely many solutions arise, even for
arbitrarily small angular velocities, in a larger, class of \textit{incomplete}
solutions that we introduce. By prescribing the transversal flux, unique
solvability of the Taylor-Couette system is recovered among rotationally
invariant incomplete solutions. Finally, we study the behavior of these
solutions as the radius of the outer disk goes to infinity, connecting our
results with the celebrated Stokes paradox.
