The problem of the effects of higher-order moments of inertia on the motion of a solid, fixed at the centre of mass and having a spherical central ellipsoid of inertia in a central Newtonian field of force is investigated. Uniform bodies of the simplest geometrical shapes (a cube, cone and cylinder) are considered. In view of the difference in the symmetries of these bodies the nature of their motions will be different. The equations of motion of a cone and a cylinder are integrated in terms of ultra-elliptic and hyperelliptic functions respectively. Sets of positions of equilibrium, permanent rotations, and regular precessions are indicated, and their branching and stability are investigated. Unlike the case when only second-order moments of inertia are taken into account, two features are determined here: 1) tow families of inclined positions with respect to equilibrium exist, and 2) for a body in the form of a cone the direct position of relative equilibrium is unstable if the vertex of the cone is situated between an attracting centre and a fixed point, and is stable otherwise, which has no analogue for permanent rotations of a body with a triaxial central ellipsoid of inertia.