  For any positive integer $k$, let $\mathcal{G}_k$ denote the set of finite
groups $G$ 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 characterize
the finite groups each of whose cubic Cayley graphs is integral. Moreover, the
class $\mathcal{G}_3$ is characterized. As an application, the classification
of $\mathcal{G}_k$ is obtained again, where $k\ge 4$.
