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Abstract: By calculating the Lyapunov characteristic exponents, an analysis was made of the stability of the secular orbital dynamics of satellites for a number of exoplanets in which the presence of an exomoon is possible. Three planets have been identified in which the existence of satellites with stable dynamics is possible: KOI-268, Kepler-1000b and Kepler-1442b. Estimates of the possible values of the exomoon orbital parameters are obtained. The dynamics of the satellites of the planets Kepler-1625b and Kepler-1708b already discovered from the analysis of observations is considered in detail. It is established that the semi-major axis of the orbit of the planet’s satellite Kepler-1625b probably has values from 10 a.u. up to 25 a.u. The solution available for the satellite of the planet Kepler-1708b (Kipping et al., 2022) corresponds to a stable satellite orbit.