Degeneracies of Periodic Solutions to the Beletsky Equation

We suggest a new method of analysis of degeneracies in families of periodic solutions to an ODE, which is based upon the application of variational equations of higher order. The equation of oscillations of a satellite in the plane of its elliptic orbit (the Beletsky equation) is considered as a model problem. We study the degeneracies of arbitrary co-dimension in the families of its $2 \pi$-periodic solutions and obtain the explicit formulas for them, which allows to localize the degeneracies with high accuracy and to give them a geometric interpretation.