A study is made of Stokes flows in which a line rotlet or stokeslet is in the presence of a circular cylinder in a viscous fluid. In contrast to the Stokes paradox for flow past an isolated cylinder, it is shown that if either type of singularity, with suitably chosen strength and location, is present, there can exist a flow which is uniform at infinity. A similar phenomenon can occur when two equal cylinders rotate with equal and opposite angular velocities, and the flow pattern is then such that there is a closed streamline enclosing both cylinders.