  We use density functional theory to study the structure and the band
structure of the monolayer FeSe deposited on the SrTiO$_3$ substrate with the
additional layer of Se between them. Top of the SrTiO$_3$ is formed by the
double TiO layer with and without oxygen vacancies. Several structures with
different arrangements of the additional Se atoms above the double TiO layer is
considered. Equilibrium structures were found and the band structures for them
were obtained. Near the $\Gamma=(0,0,0)$ point of the Brillouin zone, the hole
Fermi surface pockets persist and, additionally, an electron pocket appears.
Thus neither the presence of the additional Se layer nor the oxygen vacancies
in the double TiO layer leads to the sinking of hole bands below the Fermi
level near the $\Gamma$ point. Necessity to include the strong electronic
correlations into account is discussed.
