  A phase diagram for a 2D metal with variable carrier density has been
derived. It consists of a normal phase, where the order parameter is absent; a
so-called ``abnormal normal'' phase where this parameter is also absent but the
mean number of composite bosons (bound pairs) exceeds the mean number of free
fermions; a pseudogap phase where the absolute value of the order parameter
gradually increases but its phase is a random value, and finally a
superconducting (here Berezinskii-Kosterlitz-Thouless) phase. The
characteristic transition temperatures between these phases are found. The
chemical potential and paramagnetic susceptibility behavior as functions of the
fermion density and the temperature are also studied. An attempt is made to
qualitatively compare the resulting phase diagram with the features of
underdoped high-$T_{c}$ superconducting compounds above their critical
temperature.
