The Hamiltonian structure for a fundamental model of a tethered satellite system is constructed. The model is composed of two point masses connected by a string with no restrictions on the motions of the two masses. A certain symmetry with respect to the special orthogonal group SO(3) for such a system is observed. The classical station-keeping mode for the tethered system is found to be nothing more than the relative equilibrium corresponding to the reduction of the system by the symmetry. The microgravity forces on the two point masses are responsible for the possible configurations of the string at the so-called radial relative equilibrium. A stability analysis is performed on the basis of the reduced energy-momentum method. Criteria for stability are derived, which could find potential applications in space technology.