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For any positive integer $k$, let $\mathcal{G}_k$ denote the set of finitegroups $G$ such that all Cayley graphs ${\rm Cay}(G,S)$ are integral whenever$|S|\le k$. Est${\rm \acute{e}}$lyi and Kov${\rm \acute{a}}$cs \cite{EK14}classified $\mathcal{G}_k$ for each $k\ge 4$. In this paper, we characterizethe finite groups each of whose cubic Cayley graphs is integral. Moreover, theclass $\mathcal{G}_3$ is characterized. As an application, the classificationof $\mathcal{G}_k$ is obtained again, where $k\ge 4$.

On finite groups all of whose cubic Cayley graphs are integral