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We construct a new N=4 supersymmetric mechanics describing the motion of a particle over a $CP^n$ manifold in U(n) background gauge fields.

In this paper we construct the Lagrangian and Hamiltonian formulations of $\mathcal{N}=4$ supersymmetric systems describing the motion of an isospin particle on a conformally flat four-manifold with $SO(4)$ isometry carrying the non-Abelian field of a Belavin-Polyakov-Shvarts-Tyupkin instanton. The conformal factor can be specified to yield various particular systems, such as superconformally invariant mechanics as well as a particle on the four-sphere, the pseudosphere, or on $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{3}$. The isospin degrees of freedom arise as bosonic components of an additional fermionic $\mathcal{N}=4$ supermultiplet, whose other components are rendered auxiliary by a nonlocal redefinition. Our on-shell component action coincides with the one recently proposed in [arXiv:0912.3289].

Starting from quaternionic N=8 supersymmetric mechanics we perform a reduction over a bosonic radial variable, ending up with a nonlinear off-shell supermultiplet with three bosonic end eight fermionic physical degrees of freedom. The geometry of the bosonic sector of the most general sigma-model type action is described by an arbitrary function obeying the three dimensional Laplace equation on the sphere S{sup 3}. Among the bosonic components of this new supermultiplet there is a constant which gives rise to potential terms. After dualization of this constant one may come back to the supermultiplet with four physical bosons. However, this new supermultiplet is highly nonlinear. The geometry of the corresponding sigma-model action is briefly discussed.