  What distinguishes trivial from topological superluids in interacting
many-body systems where the number of particles is conserved? Building on a
class of integrable pairing Hamiltonians, we present a number-conserving,
interacting variation of the Kitaev model, the Richardson-Gaudin-Kitaev chain,
that remains exactly solvable for periodic and antiperiodic boundary
conditions. Our model allows us to identify fermionic parity switches that
distinctively characterize topological superconductivity in interacting
many-body systems. Although the Majorana zero-modes in this model have only a
power-law confinement, we may still define many-body Majorana operators by
tuning the flux to a fermion parity switch. We derive a closed-form expression
for an interacting topological invariant and show that the transition away from
the topological phase is of third order.
