  Particle-number projection within the Lipkin-Nogami (LN) method is applied to
the self-consistent quasiparticle random-phase approximation (SCQRPA), which is
tested in an exactly solvable multi-level pairing model. The SCQRPA equations
are numerically solved to find the energies of the ground and excited states at
various numbers $\Omega$ of doubly degenerate equidistant levels. The use of
the LN method allows one to avoid the collapse of the BCS (QRPA) to obtain the
energies of the ground and excited states as smooth functions of the
interaction parameter $G$. The comparison between results given by different
approximations such as the SCRPA, QRPA, LNQRPA, SCQRPA and LNSCQRPA is carried
out. While the use of the LN method significantly improves the agreement with
the exact results in the intermediate coupling region, we found that in the
strong coupling region the SCQRPA results are closest to the exact ones.
