Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector.

A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particle-interaction potential homogeneous of degree $α=-2$ are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems.

Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particle-interaction potential homogeneous of degree $α=-2$ are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane.

3-particle systems with a particle-interaction homogeneous potential of degree $α=-2$ is considered. A constructive procedure of reduction of the system by 2 degrees of freedom is performed. The nonintegrability of the systems is shown using the Poincare mapping.

The motion of a rigid body about its center of mass under action of gravitational moments of the central Newtonian force field is investigated. The orbit of the center of mass is proposed to be an elliptical one, the eccentricity of the orbit is equal to the one of the Mercury. The central ellipsoid of inertia of the body is arbitrary. The problem of existence of planar periodical rotations under resonance 3:2 of the Mercurian type is considered and their stability (in Liapunov) is investigated. In a case of planar perturbations the nonlinear problem of stability is solved. In a case of arbitrary perturbations of periodical rotations the stability in first (linear) approximation is investigated.