Investigating the behavior of a monopole moving in the field of another monopole we have obtained energy eigenvalue and eigenfunctions of the system. It has been demonstrated that isotropic harmonic oscillator is (n+1) (n+2)/2 fold degenerate and SU(3) is the algebra to describe this system. The energy eigenvalue of monopolonium is modified from the usual energy eigenvalue of hydrogen atom due to the magnetic charge and Bohr radius of the system is very small in comparison to atomic Bohr radius. It has also been demonstrated that Hamiltonian of the monopolonium is invariant under O(4) and degree of degeneracy is n 2 fold.

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Solving Dirac equation for a BPS monopole moving in the field of another BPS monopole in moduli space, it has been shown that spin momentum of the interacting monopole behaves as an extra energy source. The possibilities of splitting of the energy levels of the system have been explored.

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Keeping in view the partial breaking of the supersymmetry of dyons in N = 4 supersymmetric theories, which is due to the presence of a central charge in the algebra, and the mass of the BPS dyon in supersymmetric Yang-Mills theories, we have analyzed the ratios of charges of supersymmetric black holes with 1/4 of unbroken N = 4 supersymmetry and demonstrated that in spontaneous breaking of the N = 4 global supersymmetry to N = 2, the parameters of electric and magnetic Fayet-Iliopoulos terms can be considered proportional to the electric and magnetic charges of the dyonic black holes.