The integrable case in the gyrostat attitude motion in the gravitational and magnetic Earth's fields

The paper deal with a rigid body bearing a gyro, possessing an eigen magnetic moment and equipped with charged shield. Its orbit is near-Earth circle Keplerian orbit in geomagnetic equator plane. The attitude motion of the gyrostat is investigated. It is shown that four first integrals can be constructed under some conditions and therefore the solution of the problem may be reduced to the quadratures. The geometric interpretation of the gyrostat's attitude motion is given.