  We revisit the MIC-harmonic oscillator in flat space with monopole
interaction and derive the polynomial algebra satisfied by the integrals of
motion and its energy spectrum using the ad hoc recurrence approach. We
introduce a superintegrable monopole system in generalized Taub-NUT space. The
Schr\"{o}dinger equation of this model is solved in spherical coordinates in
the framework of St\"{a}ckel transformation. It is shown that wave functions of
the quantum system can be expressed in terms of the product of Laguerre and
Jacobi polynomials. We construct ladder and shift operators based on the
corresponding wave functions and obtain the recurrence formulas. By applying
these recurrence relations, we construct higher order algebraically independent
integrals of motion. We show the integrals form a polynomial algebra. We
construct the structure functions of the polynomial algebra and obtain the
degenerate energy spectra of the model.
