Spinning extensions of $D(2,1;\alpha)$ superconformal mechanics

As is known, any realization of SU(2) in the phase space of a dynamical system can be generalized to accommodate the exceptional supergroup $D(2,1;\alpha)$, which is the most general $\mathcal{N}{=}\,4$ supersymmetric extension of the conformal group in one spatial dimension. We construct novel spinning extensions of $D(2,1;\alpha)$ superconformal mechanics by adjusting the SU(2) generators associated with the relativistic spinning particle coupled to a spherically symmetric Einstein-Maxwell background. The angular sector of the full superconformal system corresponds to the orbital motion of a particle coupled to a symmetric Euler top, which represents the spin degrees of freedom. This particle moves either on the two-sphere, optionally in the external field of a Dirac monopole, or in the SU(2) group manifold. Each case is proven to be superintegrable, and explicit solutions are given.