Does the rotating space skyhook allow for a periodic motion?

Take a look at the relevant wikipedia-page to get accustomed to the concept of the skyhook. My question is whether it is possible to distribute the mass along the tether in such a way and kickstart the rotation (both the rotation around the planet and the rotation around its own center of mass) such that the tether will `essentially’ perform a periodic motion without having to burn any propellant. With an “essentially periodic" motion I mean that there must be a time $T>0$ s.t. the position arrangement of the cable at time $T$ is the same as that of time $0$ modulo a certain solid rotation $theta$ along an axis that contains the planet center. The velocity of every piece of cable-section is rotated by the same angle $theta$ relative to the velocity of the same piece of cable-section at time $0$.

I’m operating under the assumption that the planet generates a nicely radially symmetric Coulombian gravitational field and Newton’s second law governs the motion. Internal forces in the cable are local and tangent to the local tether-orientation, obey the third law and they must maintain the cable’s length locally (and globally): i.e. the cable has infinite elastic modulus.

Vague idea: maybe explore an argument similar to the one given in V. I. Arnold’s book to prove periodic solutions of double pendulum?

Bonus question: How about exploiting Poincaré recurrence? Is it true and can one prove that the tether can access merely a compact chunk of phase-space so that, by virtue of the Poincaré recurrence theorem, it will return almost to its original state after some time (thus allowing us perhaps to limit the cost of propellant-burning to maintain the station’s operation to an arbitrary small amount) ?