  Although the BCS theory of superconductivity is a well established theory, we
have shown that the phenomenology predicted by this model is much richer than
previously believed. By releasing the constraint that the attraction band is
symmetric with respect to the chemical potential of the system, we observed
that the energy gap may have more than one solution, the quasiparticle
imbalance may appear in equilibrium, and the transition between the
superconducting and the normal metal phases may be of the first order. The
temperature of the superconductor-normal metal phase transition changes with
the asymmetry of the attraction band and if we plot the phase transition
temperature vs the chemical potential, we obtain a bell shaped curve, similarly
to the superconducting dome, generally formed in high-Tc superconductors, but
also in superconductors with narrow conduction bands.
  While the pairing interaction is a microscopic characteristic of the system,
determined by the effective interactions between constituent quasiparticles,
the chemical potential is a macroscopic quantity, which can be changed by
external conditions, like doping and pressure. Furthermore, if the conduction
band of the system is narrow, then the attraction band is constrained to the
conduction band and the chemical potential is not necessary in the center, as
it happens in some of the bands in In-doped Pb$_z$Sn$_{1-z}$ and in MgB$_{2}$.
For these reasons, the constraint that the attraction band is symmetric with
respect to the chemical potential may be released.
