We show that a nonrelativistic particle in a combined field of a magnetic monopole and 1/r(2) potential reveals a hidden, partially free dynamics when the strength of the central potential and the charge-monopole coupling constant are mutually fitted to each other. In this case the system admits both a conserved Laplace`-Runge`-Lenz vector and a dynamical conformal symmetry. The supersymmetrically extended system corresponds then to a background of a self-dual or anti-self-dual dyon. It is described by a quadratically extended Lie superalgebra D(2, 1;alpha) with alpha = 1/2, in which the bosonic set of generators is enlarged by a generalized Laplace`-Runge`-Lenz vector and its dynamical integral counterpart related to Galilei symmetry, as well as by the chiral Z(2)-grading operator. The odd part of the nonlinear superalgebra comprises a complete set of 24 = 2 x 3 x 4 fermionic generators. Here a usual duplication comes from the Z(2)-grading structure; the second factor can be associated with a triad of scalar integrals-the Hamiltonian, the generator of special conformal transformations, and the squared total angular momentum vector, while the quadruplication is generated by a chiral spin vector integral which exits due to the (anti-)self-dual nature of the electromagnetic background.