Geometry and integrability in $\mathcal{N}=8$ supersymmetric mechanics

We construct the $\mathcal{N}=8$ supersymmetric mechanics with potential term whose configuration space is the special K"ahler manifold of rigid type and show that it can be viewed as the K"ahler counterpart of $\mathcal{N}=4$ mechanics related to “curved WDVV equations”. Then, we consider the special case of the supersymmetric mechanics with the non-zero potential term defined on the family of $U(1)$-invariant one-(complex)dimensional special K"ahler metrics. The bosonic parts of these systems include superintegrable deformations of perturbed two-dimensional oscillator and Coulomb systems.