These motions of a rigid body have firstly been studied by G. Darboux in 1895. The are the simplest non-trivial rational motions, having algebraic order 2. Darboux motions are generated by composing a planar elliptic motion with a harmonic oscillation. All trajectories are ellipses, i.e., rational curves of algebraic order 2. As already observed by Darboux, these motions are the most general spatial motions with only planar trajectories. The figure shows a Darboux motion and its kinematical generation.
This is an animated version of the image:
A general construction and classification of rational motions of arbitrary order has been presented in the article entitled "Über zwangläufige rationale Bewegungsvorgänge"