= Solution
Choose the vertical plane through the rod's axis. Reflection in that plane leaves the horizontal rod, the vertical gravitational force, and the <Stokes flow>[Stokes equations] unchanged, but reverses any velocity normal to the plane. A second reflection in the vertical plane perpendicular to the rod reverses velocity along the rod. By uniqueness of <Stokes flow>, both horizontal velocity components vanish. The rod therefore translates vertically.
Equivalently, the <hydrodynamic resistance matrix> of an axisymmetric rod has one coefficient parallel to its axis and another perpendicular to it. A vertical force on a horizontal rod lies entirely in the perpendicular eigenspace, so its velocity is vertical.
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