Non complete integrability of a magnetic satellite in circular orbit

We consider the motion of a rigid body (for example a satellite) on a circular orbit around a fixed gravitational and magnetic center. We study the non complete meromorphic integrability of the equations of motion which depend on parameters linked to the inertia tensor of the satellite and to the magnetic field. Using tools from computer algebra we apply a criterion deduced from J.-J. Morales and J.-P. Ramis theorem which relies on the differential Galois group of a linear differential system, called normal variational system. With this criterion, we establish non complete integrability for the magnetic satellite with axial symmetry, except for a particular family F already found in [11], and for the satellite without axial symmetry. In the case of the axial symmetry, we discuss the family F using higher order variational equations ([14]) and also prove non complete integrability.