  Nuclear level density $\rho(E,A)$ is derived for a nuclear system with a
given energy $E$ and particle number $A$ within the mean-field semiclassical
periodic-orbit theory beyond the saddle-point method, obtaining $~~\rho \propto
I_\nu(S)/S^\nu$, where $I_\nu(S)$ is the modified Bessel function of the
entropy $S$. Within the micro-macrocanonical approximation (MMA), for a small
thermal excitation energy, $U$, with respect to rotational excitations, $E_{\rm
rot}$, one obtains $\nu=3/2$ for $\rho(E,A)$. In the case of larger excitation
energy $U$ but smaller the neutron separation energy, one finds a larger value
of $\nu=5/2$. A role of the fixed spin variable for rotating nuclei is
discussed. The MMA level density $\rho$ reaches the well-known grand-canonical
ensemble limit (Fermi gas asymptotics) for large $S$ related to large
excitation energies, and the finite micro-canonical limit for small
combinatorical entropy $S$ at low excitation energies (constant "temperature"
model). Fitting the MMA $\rho(E,A)$ to the experimental data and taking into
account shell and, qualitatively, pairing effects for low excitation energies,
one obtains the inverse level density parameter $K$, which differs essentially
from that deduced from data on neutron resonances.
