For a connected graph $G$ an edge set $S$ is a restricted edge cut if $G-S$ is disconnected and there is no isolated vertices in $G-S$. The number of edges in a restricted edge cut of minimum cardinality is known as the restricted edge connectivity of $G$, denoted by $\lambda' (G)$. A restricted edge connected graph is minimally restricted $2$-edge connected if $\lambda' (G) = 2$ and $\lambda'(G-e) =1$, for every edge $e\in E(G).$ In this work we study some structural properties of minimally restricted $2$-edge connected graphs. Moreover, we give a necessary and sufficient condition for a $\lambda'$-connected graph $G$ with vertex connectivity $\kappa(G)=2$ to be minimally restricted $2$-edge connected.