The paper considers a limited formulation of the motion of a gyrostat along a Keplerian circular orbit in a central Newtonian field of forces with a thin homogeneous elastic rod clamped by one end in its carrier. A one-parameter family of nontrivial relative equilibriums of the system (the state of rest relative the orbital coordinate system, except for the flywheel when the rod is deformed) is presented, defined by the arbitrary position of the unit vector of the local vertical in the plane perpendicular to the initial unstrained rod axis. Sufficient conditions for Liapunov stability of the nontrivial relative equilibriums of that family are determined. The investigation was done under some simplifying assumptions. In particular it was assumed that the displacement vector of an arbitrary point of the rod axis under elastic deformation can be represented by an infinite series (without a priori truncation) of expansion on given set of functions depending on the spatial coordinates of point multiplied by unknown time-dependent coefficients.
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