  Several approximations are tested by calculating the ground-state energy and
the energy of the first excited $0^{+}$ state using an exactly solvable model
with two symmetric levels interacting via a pairing force. They are the BCS
approximation (BCS), Lipkin - Nogami (LN) method, random-phase approximation
(RPA), quasiparticle RPA (QRPA), the renormalized RPA (RRPA), and renormalized
QRPA (RQRPA). It is shown that, in the strong-coupling regime, the QRPA which
neglects the scattering term of the model Hamiltonian offers the best fit to
the exact solutions. A recipe is proposed using the RRPA and RQRPA in
combination with the pairing gap given by the LN method. Applying this recipe,
it is shown that the normal-superfluid phase transition is avoided, and a
reasonably good description for both of the ground-state energy and the energy
of the first excited $0^{+}$ state is achieved.
