One-axis equilibrium orientations to an attracting center of a symmetric prolate orbital gyrostat with an elasctic beam

We study the motion of a symmetric prolate stationary gyrostat along a Keplerian circular orbit in a Newtonian central field of forces in the restricted formulation. An elastic beam is clamped by one end in the body of the gyrostat along its axis of symmetry. The beam has a point mass at the free end. The inextensible elastic beam (which is, for simplicity, of constant circular cross-section) performs infinitesimal space vibrations in the process of the motion of the system. Moreover, we neglect the terms nonlinear with repect to the displacements of the points of the beam in the tensor of inertia of the system. We consider the following (so-called semi-inverse) problem: Under what kinetic moment of the gyrostat among its relative equilibria (the states of rest in the orbital coordinate system) is an arbitrary coordinate axis defined in the coordinate system associated with the gyrostat collinear to the local vertical? In the discretization of the problem, we give the values of the Lagrange coordinates defining the deformation of the beam in these equilibria and the value of the gyrostatic moment guaranteeing the presence of the equilibrium.