The project is devoted to a semi-analytical and numerical study of dynamics in quasi-satellite regime of motion in the planetary three-body problem. The main tool in this study is a construction of a quasi-integrable approximation of the problem with a numerical study of dynamics in the framework of this approximation. Three-body problem is a problem of description of motion of three point masses (bodies) under the action of their mutual gravitational attraction. Planetary (or astronomical) three body problem is a version of the general three-body problem when mass of one body (a star) is much larger than masses of two other bodies (planets). This version of the three-body problem is used to describe motion of planets around a star or motion of satellites around a planet. A quasi-satellite regime of motion in such a problem is a regime in which two planets stay near each other for a long time, but remain outside of the corresponding Hill's spheres (regions in space where planets can hold their satellites).<br><br>Detailed description of quasi-satellite orbits for the asteroidal (or restricted planetary) three-body problem is given in [1] (asteroidal problem is a version of the planetary problem when one of the planets has a negligible mass). A particular case of the planetary problem with nearly-circular orbits of planets is studied in [2]. The project is aimed to study a general case using an approach based on the theory of slow-fast Hamiltonian systems similarly to [1]. This approach leads to a quasi-integrable approximation of the problem. Quasi-integrablity means that the phase space of the approximate system is glued of several pieces. In each of these pieces the system is integrable. Passage between pieces leads to a probabilistic scattering described in [3]. The approximate first integrals of motion have a complicated structure and can't be given in an explicit form. So, numerical study of the structure of joint level manifolds of these first integrals is required for a complete description of dynamics.