The monograph describes modern mathematical methods of qualitative analysis of dynamical systems in relation to the classical problem of rotation of a solid body with a fixed point. The problems considered are grouped around three related problems: the existence of unambiguous analytical integrals, periodic solutions, small denominators. These problems occupy one of the central places in classical mechanics. The first edition was published in 1980. and has long been a bibliographic rarity. The new edition includes work in . In. Kozlov, devoted to the study of the Duffing equations