We consider the ordinary differential equation of the second order describing oscillations of a satellite in a plane of its elliptical orbit. The equation contains two parameters: $e$ and $\mu$. It is regular for $0\leq e<1$ and singular for $e=1$. We have computed five families of symmetric (odd) periodic solutions for $|\mu|\leq20$ and for $e=0,0.1,0.5,0.9,0.99,0.999$. We have also computed the corresponding values of the trace characterising their stability. For $e>0.9$ we use the regularization by means of the eccentric anomaly. Results are given in figures. They show that for $e\to1$ these families tend to some limiting positions.