  We demonstrate that the well-known $(k\uparrow, -k\downarrow)$
Bardeen-Cooper-Schrieffer interaction, when considered in real space, is
equivalent to an infinite-range Penson-Kolb pairing mechanism coexisting with
an attractive Hubbard term. Driven by this discovery and aiming at exploring
the conduction properties, we investigate the dynamics of fermionic particles
confined in a ring-shaped lattice. We assume that fermions are simultaneously
influenced by the pairing interaction and by an Aharonov-Bohm electromagnetic
phase, which is incorporated into the model in a highly non-trivial manner.
Remarkably, the aforementioned model shows Richardson integrability for both
integer and half-integer values of the applied magnetic flux $\Phi/\Phi_0$,
thus permitting the exact solution of a genuine many-body problem. We discuss
the ground state properties of both two-particle and many-particle systems,
drawing comparisons with results from the attractive Hubbard model. Our
approach combines exact diagonalization, density matrix renormalization group
techniques, and numerical solution of the Richardson equations. This
comprehensive analysis allows us to study various key metrics, including the
system's conductivity as a function of the interaction strength. In this way,
the BCS-BEC transition is investigated in a continuous manner, thus permitting
to shed light on fundamental aspects of superconducting pairing. Our findings
can be experimentally tested in a condensed matter context or, with greater
level of control, using atomtronics platforms.
