Astronomical and Astrophysical T ransactions (AApT r), 2018, Vol. 30, Issue 3, pp. 359–362, ISSN 1055-6796, Photocopying permitted by license only, c Cambridge Scientific Publishers Formation and stationary state of a spindle-shaped collisionless gravitating system Zafar Turakulov ∗ Ulugh Bek Astronomy Institute (UBAI), Astronomicheskaya 33, Tashkent 700052, Uzbekistan Received 30 January, 2017 Possible scenario of formation and analytic solution of Vlasov equation for the final state of a self-gravitating collisionless system are proposed. The system is assumed to possess axial symmetry and zero total kinetic energy in the initial state. It is shown that under these initial conditions evolution of the velocity distribution goes a special way and after all finalizes with formation of phase density of special form. It turns out that for this form of the phase density the Vlasov equation is solvable. A particular solution is obtained which describes a spindle-shaped cluster of gravitating particles. Keywords: Axisymmetric collisionless system, analytic solution 1 Introduction A collisionless system whose particles stay at rest at some initial moment of time, starts to change in shape and density under the action of gravitational forces and after some period of evolution, all changes slow down as the system approaches its equilibrium state. Symmetry of the initial state, particularly, axial symmetry prede- termines some properties of the equilibrium state which theoretically is the final of evolution of the system. In this work we discuss the formation of the final state of an initially cold axially-symmetric collisionless self-gravitating system. The goal is to obtain an analytic solution of the Vlasov equation for the state formed under these conditions. In the following considerations we assume that there are no instabili- ties which break the axial symmetry, hence its gravitational potential remains axially symmetric. It is natural to consider the system in round cylinder coordinates {z,ρ,ϕ}. If the motion of each particle of the system starts from the resting state, then accel- erates strictly in the half-plane of constant azimuthal angle ϕ and ϕ-component of its velocity remains zero forever. Then, during the whole period of evolution and in the final state all particles of the system move only in half-planes ϕ = const and * Email: zafarturakulov@gmail.com