  In this paper we analyze computational properties of the Diophantine problem
(and its search variant) for spherical equations $\prod_{i=1}^m z_i^{-1} c_i
z_i = 1$ (and its variants) over the class of finite metabelian groups
$G_{p,n}=\mathbb{Z}_p^n \rtimes \mathbb{Z}_p^\ast$, where $n\in\mathbb{N}$ and
$p$ is prime. We prove that the problem of finding solutions for certain
constrained spherical equations is computationally hard on average (assuming
that some lattice approximation problem is hard in the worst case).
