  A self-gravitating sphere of polytropic gas (polytrope) is considered. The
system of equations describing radial motions of this sphere in Lagrangian
variables reduces to the only nonlinear PDE of the second order in both
variables (Lagrangian coordinate and time). The linearization of this PDE leads
to the well-known Eddington's equation of the standard model. The case of no
energy exchange between the polytrope and the outer medium is considered, that
is, polytrope's motions are adiabatic. If gamma (a ratio of the specific heats
of the gas) is 4/3 than PDE obtained allows the separation of variables. There
exist two types of solutions of the problem both describing limitless expansion
without shock wave formation. The first one is an expansion with positive total
energy, and the second one is an expansion with zero total energy. The second
solution is of an astrophysical interest. It describes the permanently
retarding expansion that, perhaps, is akin to a born of a red giant. The
stellar density in this case concentrates to the centre of the star stronger
than the density of the stationary star with the same gamma.
