Extended Heine-Stieltjes polynomials related to the isovector pairing model

New polynomials related to the mean-field plus isovector pairing model are constructed based on the Stieltjes correspondence. It is shown that there is an one-to-one correspondence between zeros of the two related extended Heine-Stieltjes polynomials satisfying coupled differential equations and the solutions of the Bethe ansatz equations for the model. Similar to the standard pairing among like valence nucleons, an electrostatic interpretation of the location of zeros of the two polynomials is provided. As examples of the solution, the two polynomials related to the solution for three pairs over $$j=1/2,\,3/2,\,5/2$$ orbits within the O(5) seniority-zero subspace are provided explicitly, which shows that the number of possible configurations of equilibrium positions of the charges in the two separate systems equals exactly to the number of energy levels in the mean-field plus isovector pairing model.