  We present here a microscopic two-band model based on the structure of
energetic levels of holes in $\mathrm{CuO}_{2}$ conducting layers of cuprates.
We prove that two energetically near-lying interacting bands can explain the
electron-hole asymmetry. Indeed, we rigorously analyze the phase diagram of the
model and show that the critical temperatures for fermion densities below
half-filling can manifest a very different behavior as compared to the case of
densities above half-filling. This fact results from the inter-band interaction
and intra-band Coulomb repulsion in interplay with thermal fluctuations between
two energetic levels. So, if the energy difference between bands is too big
then the asymmetry disappears. Moreover, the critical temperature turns out to
be a non-monotonic function of the fermion density and the phase diagram of our
model shows "superconducting domes" as in high-$T_{c}$ cuprate superconductors.
This explains why the maximal critical temperature is attained at donor
densities away from the maximal one. Outside the superconducting phase and for
fermion densities near half-filling the thermodynamics governed by our
Hamiltonian corresponds, as in real high-$T_c$ materials, to a Mott-insulating
phase. The nature of the inter-band interaction can be electrostatic (screened
Coulomb interaction), magnetic (for instance some Heisenberg-type one-site
spin-spin interaction), or a mixture of both. If the inter-band interaction is
predominately magnetic then - additionally to the electron-hole asymmetry - we
observe a reentering behavior meaning that the superconducting phase can only
occur in a finite interval of temperatures. This phenomenon is rather rare, but
has also been observed in the so-called magnetic superconductors.
