  Recently, it has been possible to construct single-electron transistors to
study electronic properties, including superconductivity, in metallic grains of
nanometer size. Among several theoretical results are suppression of
superconductivity with decreasing grain size and parity effect, i.e. dependence
on the parity of the number of electrons on the grain. We study how these
results are affected by degeneracy of energy levels. In addition to the
time-reversal symmetry, for certain energy spectra and more generally for
lattice symmetries, energy levels are strongly degenerate near the Fermi
energy. For a parabolic dispersion, degeneracy d is of the order of k_FL while
the typical distance between the levels is of the order of epsilon_F/(k_FL)^2
where k_F and epsilon_F are the Fermi wave vector and energy, respectively, and
L is the particle size. First, using an exact solution method for BCS
Hamiltonian with finite number of energy levels, for the well studied
non-degenerate case we find a new feature. In that case, parity effect exhibits
a minimum instead of a monotonic behavior. For d-fold degenerate states we find
that the ratio of two successive parity effect parameters Delta_p is nearly
1+1/d. Our numerical solutions for the exact ground state energy of negative U
Hubbard model on a cubic cluster also give very similar results. Hence we
conclude that parity effect is a general property of small Fermi systems with
attractive interaction, and it is closely related to degeneracy of energy
levels.
