  Static spherically symmetric solution of the Einstein's equations is found
representing averaged properties of an infinite self-gravitating gas in the
dynamical equilibrium. It depends upon three parameters: the core radius, the
relativistic factor $0<z<1$, defining the density and the mass, and one
structural parameter. In the relativistic limit, $z\to 1$, an open nucleus of
focused gravitational fields develops with the size being a small fraction of
the core radius whereas the outskirts of the self-gravitating gas remains
non-relativistic. Inside the nucleus the self-gravitating gas is described by a
universal perfect fluid with the relativistic one-dimensional equation of
state. At $z=1$, the space and time develops a point like singularity at the
center. A characteristic property of the nucleus is incompressibility. New
particles added to the self-gravitating gas bounce back into the outskirts
rather than fall into the center.
