The motion of a gyrostat, regarded as a rigid body, in a circular Kepler orbit in a central Newtonian force field is investigated in a limited formulation. A uniformly rotating statically and dynamically balanced flywheel is situated in the rigid body. A uniform elastic element, which, during the motion of the system, is subjected to small deformations, is rigidly connected to the rigid body-gyrostat body. The problem is discretized without truncating the corresponding infinite series, based on a modal analysis or using a certain specified system of functions, for example, of the assumed forms of the oscillations, which depend on the spatial coordinates and which satisfy appropriate boundary-value problems of the linear theory of elasticity. The elastic element is specified in more detail (a rod, plate, etc.), as well as its mass and stiffness characteristics and the form of the fastening, and the choice of the system of functions is determined. Non-trivial relative equilibria of the system (the state of rest with respect to an orbital system of coordinates when the elastic element is deformed) is sought approximately on the basis of a converging iteration method, described previously. It is shown, using Routh's theorem, that by an appropriate choice of the gyrostatic moment and when certain conditions, imposed on the system parameters are satisfied, one can stabilize these equilibria (ensure that they are stable).