On closed distance magic circulants of valency up to $5$

Let $\mathrm{\Gamma }=(V,E)$ be a graph of order $n$. A {\em closed distance magic labeling} of $\mathrm{\Gamma }$ is a bijection $\mathrm{\ell }:V\to \{1,2,\dots ,n\}$ for which there exists a positive integer $r$ such that ${\sum }_{x\in N[u]}\mathrm{\ell }(x)=r$ for all vertices $u\in V$, where $N[u]$ is the closed neighborhood of $u$. A graph is said to be {\em closed distance magic} if it admits a closed distance magic labeling. In this paper, we classify all connected closed distance magic circulants with valency at most $5$, that is, Cayley graphs $\mathrm{Cay}({\mathbb{Z}}_n;S)$ where $\mathrm{\mid }S\mathrm{\mid }\le 5$ and $S$ generates ${\mathbb{Z}}_n$.