  A fermion realization of the compact symplectic sp(4) algebra provides a
natural framework for studying isovector pairing correlations in nuclei. While
these correlations manifest themselves most clearly in the binding energies of
0^+ ground states, they also have a large effect on the energies of excited
states, including especially excited 0^+ states. In this article we consider
non-deformed as well as deformed algebraic descriptions of pairing through the
reductions of sp_{(q)}(4) to different realizations of u_{(q)}(2) for single-j
and multi-j orbitals. The model yields a classification scheme for completely
paired 0^{+} states of even-even and odd-odd nuclei in the 1d_{3/2}, 1f_{7/2},
and 1f_{5/2}2p_{1/2}2p_{3/2}1g_{9/2} shells. Phenomenological non-deformed and
deformed isospin-breaking Hamiltonians are expressed in terms of the generators
of the dynamical symmetry groups Sp(4) and Sp_{q}(4). These Hamiltonians are
related to the most general microscopic pairing problem, including isovector
pairing and isoscalar proton-neutron interaction along with non-linear
interaction in the deformed extension. In both the non-deformed and deformed
cases the eigenvalues of the Hamiltonian are fit to the relevant Coulomb
corrected experimental 0^{+} energies and this, in turn, allows us to estimate
the interaction strength parameters, to investigate isovector-pairing
properties and symmetries breaking, and to predict the corresponding energies.
While the non-deformed theory yields results that are comparable to other
theories for light nuclei, the deformed extension, which takes into account
higher-order interactions between the particles, gives a better fit to the
data. The multi-shell applications of the model provide for reasonable
predictions of energies of exotic nuclei.
