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The nonrelativistic motion of a charged particle around a dyon in (9+1)spacetime is known as the nine-dimensional MICZ-Kepler problem. This problemhas been solved exactly by the variables-separation method in three differentcoordinate systems, spherical, parabolic, and prolate spheroidal. In thepresent study, we establish a relationship between the variable separation andthe algebraic structure of SO(10) symmetry. Each of the spherical, parabolic,or prolate spheroidal bases is proved to be a set of eigenfunctions of acorresponding nonuplet of algebraically-independent integrals of motion. Thisfinding also helps us establish connections between the bases by the algebraicmethod. This connection, in turn, allows calculating complicated integrals ofconfluent Heun, generalized Laguerre, and generalized Jacobi polynomials, whichare important in physics and analytics.

Algebraic structure underlying spherical, parabolic and prolate spheroidal bases of the nine-dimensional MICZ-Kepler problem