The critical subfamilies of the family $K_0$ of periodic solutions to the equation of oscillations of a satellite

We consider the ordinary differential equation of the second order describing oscillations of a satellite in a plane of its elliptic orbit. There exists an infinite number of two-parameter families $K_i$, $i=0,1,\dots$, of odd $2\pi$-periodic solutions to the equation. The domains of stability in every family $K_i$ are bounded by one-parameter subfamilies of critical solutions, which have the trace $\mathrm{Tr}=\pm2$. This study gives the complete description of all critical subfamilies of the family $K_0$. It is shown that domains of stability have a fractal (or self-similar) structure.