Rotations of a dumbbell equipped with ‘the leier constraint’

We consider a special space tethered system consisting of a dumbbell-shaped rigid body and a particle. The particle coast along on the cable. The cable ends are placed in the dumbbell endpoints. We call such system ‘the system with leier constraint (the Dutch term ‘leier’ means the rope with both fixed ends). We assume that the system mass center moves along the circular orbit in the Newtonian Central Force Field. We study the dumbbell's relative motion caused by the particle of small mass in the orbital frame of reference. We deduce a sufficient condition for librations of the dumbbell about its stable equilibrium. We find a family of the dumbbell's asymptotic motions tending to librations about unstable equilibrium. The surface of such asymptotic motions is an interstream separating the areas of the dumbbell's right-hand and left-hand rotations. We deduce an equation of this surface