  It was recently shown that the BCS formalism leads to several solutions for
the energy gap and the equilibrium quasiparticle distribution, with a phase
transition temperature which depends on the position of the chemical potential
within the attraction band (the attraction band AB is defined as the
single-particle energy interval in which the pairing interaction is
manifested). Moreover, in some cases, the phase transition may be of the first,
not of the second order. Here I will find two sets of solutions for any
temperature below the phase transition temperature. I will also show that, when
the AB is symmetric with respect to the chemical potential (the textbook BCS
problem) there are still two solutions, with different energy gaps: one
solution is the typical (textbook) BCS solution, whereas the other one has a
smaller energy gap and non-zero quasiparticle populations down to zero
temperature. At zero temperature, the energy gap corresponding to the second
solution is one third of the typical BCS solution.
