Tidal interactions between Planet and its satellites are known to be the main
phenomena, which are determining the orbital evolution of the satellites. The modern
ansatz in the theory of tidal dissipation in Saturn was developed previously by the inter-
national team of scientists from various countries in the field of celestial mechanics. Our
applying to the theory of tidal dissipation concerns the investigating of the system of ODE-
equations (ordinary differential equations) that govern the orbital evolution of the satel-
lites; such an extremely non-linear system of 2 ordinary differential equations describes the
mutual internal dynamics for the eccentricity of the orbit along with involving the semi-
major axis of the proper satellite into such a monstrous equations. In our derivation, we
have presented the elegant analytical solutions to the system above; so, the motivation of
our ansatz is to transform the previously presented system of equations to the convenient
form, in which the minimum of numerical calculations are required to obtain the final
solutions. Preferably, it should be the analytical solutions; we have presented the solution
as a set of quasi-periodic cycles via re-inversing of the proper ultra-elliptical integral. It
means a quasi-periodic character of the evolution of the eccentricity, of the semi-major
axis for the satellite orbit as well as of the quasi-periodic character of the tidal dissipation
in the Planet.

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• About Tidal Evolution of Quasi-Periodic Orbits of Satellites