We consider a steady viscous incompressible fluid occupying the exterior of an infinite cylinder. Under the action of a given external force, the motion of the fluid is described by the classical Navier–Stokes equations. Our focus is on a class of solutions in which the velocity field is vertically uniform and at rest at infinity. This configuration includes the 2D exterior problem for the steady Navier-Stokes equations known to have characteristic difficulties. We will report that for a given force with suitable decay, one can find a solution in the above class. The proof requires symmetry on domains, but not on forces. We will also touch on related problems in other contexts and some open problems. Part of this talk is based on joint work with Ryoma Horiuchi (Kobe University).