On distance-regular Cayley graphs over abelian groups of rank $2$

In 2007, Miklavi{c} and Poto{c}nik proposed the problem of characterizing distance-regular Cayley graphs over specified groups, which can be viewed as a natural extension of the problem of characterizing strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, we consider the Miklavi{c}-Poto{c}nik problem for abelian groups of rank $2$. More specifically, we determine all distance-regular Cayley graphs over the group Zₙ⊕ Zₚ, where p is an odd prime. Our proof use some new tools such as polynomial addition set, Desarguesian affine plane, and duality of Schur rings over abelian groups.