Localization of Degeneracies in Families of Periodic Solutions to an ODE and their Regularization

We suggest a new method of localization of degeneracies in families of periodic solutions to an ODE and their regularization, which is based upon the application of variational equations of higher order. For the equation of oscillations of a satellite in the plane of its elliptic orbit (the Beletskiy equation) We study the degeneracies of arbitrary co-dimension in the families of its 2$\pi$-periodic solutions. For all known degeneracies in these families which exist when |e|<1 the explicit formulae are given, which allow to localize them with high accuracy. It is shown that the families of odd critical 2$\pi$-periodic solutions with trace Tr= ±2 are the special case of so called extremal families of solutions to the Beletskiy equation, i.e. solutions on which parameter e and $\mu$ attain their extremal values.