THE FIGURES OF EQUILIBRIUM OF BARRED STELLAR-SYSTEMS WITH A NEEDLE-SHAPED ELLIPSOID OF VELOCITIES - THE GENERAL PROBLEM

Three classes of new self-consistent phase models of stellar systems with a square-law potential are discovered; these models describe the figures of equilibrium of gravitation collisionlese ellipsoids. The new figures of equilibrium have the following properties; 1) they have an (oblique) rotation, i.e. their rotation axes in a general case do not coincide with any axis of inertia of the material ellipsoid; 2) in the space of random velocities, these figures are represented by one-dimensional straight line sections ("needles"), implying the existence of linear relations between stellar velocity vectors in a given point of the configuration space; 3) the figures rotate with such an angular velocity, that, in a general case, they have no <<special>> axis, in each point of which the centrifugal force and gravity are just balanced. In the phase space the models are represented by four-dimensional (in a particular case - by three-dimensional) ellipsoids. It is proved that the majority of stars do not touch the elliptical boundary and that the contact is possible only for stars from the ends of the velocity needles. Then, a general problem for the models with a needle-shaped ellipsoid of velocities is considered. The differential equations of motion of an individual star are solved, and the four first integrals of this motion are found.