  An analytic solution has been found in the Roche approximation for the
axially symmetric structure of a hydrostatically equilibrium atmosphere of a
neutron star produced by collapse. A hydrodynamic (quasione-dimensional) model
for the collapse of a rotating iron core in a massive star gives rise to a
heterogeneous rotating protoneutron star with an extended atmosphere composed
of matter from the outer part of the iron core with differential rotation
(Imshennik and Nadyozhin, 1992). The equation of state of a completely
degenerate iron gas with an arbitrary degree of relativity is taken for the
atmospheric matter. We construct a family of toroidal model atmospheres with
total masses $M \approx 0.1 \div 0.2 M_{\odot}$ and total angular momenta $J
\approx (1 \div 5.5) \cdot 10^{49} erg \cdot s$, which are acceptable for the
outer part of the collapsed iron core, in accordance with the hydrodynamic
model, as a function of constant parameters $\omega_{0} and r_{0}$ of the
specified differential rotation law $\Omega =
\omega_{0}\exp{\Big[-\frac{(r\sin{\Theta})^{2}}{r_{0}^{2}}\Big]}$ in spherical
coordinates. The assumed rotation law is also qualitatively consistent with the
hydrodynamic model for the collapse of an iron core.
