  Background: Computationally tractable models of atomic nuclei is a long-time
goal of nuclear structure physics. A flexible framework which easily includes
excited states and many-body correlations is the configuration-interaction
shell model (SM), but the exponential growth of the basis means one needs an
efficient truncation scheme, ideally one that includes both deformation and
pairing correlations.
  Purpose: We propose an efficient truncation scheme of the SM: starting from a
pair condensate variationally defined by Hartree-Fock single-particle states
and the particle-number conserved Bardeen-Cooper-Schrieffer (NBCS)
approximation, we carry out projection of states with good angular momentum.
  Methods: After generating Hartree-Fock single-particle states with Kramers
degeneracy in a SM space, we optimize the pair amplitudes in the NBCS by
minimizing the energy, and then use linear algebra projection (LAP) of states
with good angular momentum. Both NBCS and LAP are computationally fast.
  Results: Our calculations yield good agreement with full
configuration-interaction SM calculations for low-lying states of transitional
and rotational nuclei with axially symmetric and triaxial deformation in
medium- and heavy-mass regions: $^{44,46,48}$Ti, $^{48,50}$Cr, $^{52}$Fe,
$^{60,62,64}$Zn, $^{66,68}$Ge, $^{68}$Se, and $^{108,110}$Xe. We predict
low-lying states of $^{112-114}\textrm{Ba}$ and $^{116-120}\textrm{Ce}$, nuclei
difficult to reach by large-scale SM calculations.
  Conclusions: Both pair correlation and the configuration mixing between
different intrinsic states play a key role in reproducing collectivity and
shape coexistence, demonstrating the utility of this truncation scheme of the
SM to study transitional and deformed nuclei.
