The modified system of equations described motions of a rigid body

In the case $y_0=z_0=0$ the system of 6 autonomous ordinary differential equations of the first order, describing motions of a rigid body with a fixed point, is reduced to a system of 2 nonautonomous ODE of the second order, having 2 first integrals; the reduction is made by means of a coordinate change (§ 1). In § 2 we describe a method of computation of all power expansions of solutions of those ODE systems. It uses the truncated systems and is based on the Power Geometry. In § 3 we compute supports, polyhedrons, faces and their normal cones for both equations and for both first integrals of the reduced ODE system. We show that in the generic case it is enough to study only 4 truncated systems among 18.