  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.
