In this work, we study the full set of eigenstates of a px + ipy topological superconductor coupled to a particle bath which can be described in terms of an integrable Hamiltonian of the Richardson-Gaudin class. The results derived in this work also characterise the behaviour of an anisotropic XXZ central spin model in a external magnetic field since both types of Hamiltonian are know to share the exact same conserved quantities making them formally equivalent.We show how by ramping the coupling strength (or equivalently the magnetic field acting in the z-direction on the central spin), each individual eigenstate undergoes a sequence of gain/loss of excitations when crossing the specific values known as Read-Green points. These features are shown to be completely predictable, for every one of the 2 N eigenstates, using only two integers obtainable easily from the zero-coupling configuration which defines the eigenstate in question.These results provide a complete map of the particle-number sectors (superconductor) or magnetisation sectors (central spin) involved in the large number of level-crossings which occur in these systems at the Read-Green points. It further allows us to define quenching protocols which could create states with remarkably large excitation-number fluctuations.