Properties of certain functions of celestial mechanics

In [1] the planar motion of a satellite in an elliptic orbit by the Krylov–Bogolyubov asymptotic method is studied. In particular, oscillations of a satellite in an absolute coordinate system are considered. In this case, the terms of the first, second, and fourth orders in the small parameter are trivially equal to zero in the averaged system. In this article, the proofs of nontrivial statements are given for the above-mentioned preprint, and viz., the terms of the third order are also zero and certain coefficients in terms of the fifth order are equal to each other.