  The problem of constructing or characterizing strongly regular Cayley graphs
(or equivalently, regular partial difference sets) has garnered significant
attention over the past half-century. In [European J. Combin. 24 (2023)
777--784], Miklavi\v{c} and Poto\v{c}nik expanded upon this field, achieving a
complete characterization of distance-regular Cayley graphs over cyclic groups
by employing the method of Schur rings. In [J. Combin. Theory Ser. B 97 (2007)
14--33], they formally proposed the problem of characterizing distance-regular
Cayley graphs for any given class of groups. In this context, abelian groups
are of particular importance, as a considerable number of distance-regular
graphs with classical parameters are precisely Cayley graphs over abelian
groups. In this paper, we address the Miklavi\v{c}-Poto\v{c}nik problem for
abelian groups of rank $2$. More specifically, we identify all distance-regular
Cayley graphs over the group $\mathbb{Z}_n \oplus \mathbb{Z}_p$, where $p$ is
an odd prime. Our proof utilizes novel tools such as polynomial addition sets,
Desarguesian affine planes, and the duality of Schur rings over abelian groups.
