Local expansions of modified motions of a rigid body

Here we continue the study of power expansions of solutions to the modified system of equations, describing motions of a rigid body with a fixed point, begun in the preprint no. 49 (2001). Here we correct its misprints and introduce new parameters (§ 1$'$). In § 4 we show how the general methods of Power Geometry must be applied to that problem with two equations of motions and with two first integrals. We have studied all power expansions of solutions, corresponding to the vertex $\Gamma_1^{(0)}$ (in § 5) and to the edge $\Gamma_2^{(1)}$ (in § 6) of the polyhedron of equations of motions. We have found new cases of the integrability with respect to the fractional powers of the independent variable.