The stability of steady rotation of a closed flexible filament, closed in a ring around an attracting centre is investigated as a function of the filament extensibility. Characteristics ensuring the stability of rotation of the ring are determined. These do not correspond to Hooke's law. The results are used to investigate the stability of steady rotation of a large number of artificial satellites connected in series along the orbit in a ring by weightless cables of variable length. A wide class of control laws is indicated for the tension of the connecting cables that ensure the stability of the satellite ring. An analogy in the dynamics of meteor and satellite rings is described a model of a meteor ring formed by the equivalent flexible ring having a defined law of extensibility is proposed.