This paper deals with a gyrostat satellite in a weakly elliptical near-Earth orbit with small inclination. The gyrostat is supplied with flywheel and possesses electrostatic charge and intrinsic magnetic moment. The gyrostat attitude motion under the action of Lorentz, magnetic and gravity-gradient torques is studied. It is shown that the system of differential equations for attitude motion of gyrostat with triaxial ellipsoid of inertia is reversible with two fixed sets. In the case of a spherical ellipsoid of inertia, the third fixed set occurs. It is revealed that two of these three sets correspond to the respective sets of the generating problem, that is, in the case of a circular equatorial orbit. It is made clear what the families of symmetric oscillations of the generating problem bifurcate and give rise to isolated oscillations. Thus, all periodic oscillations are found for gyrostat with equal moments of inertia on weakly elliptic orbit with small inclination.