  An $S$-ring (Schur ring) is called separable with respect to a class of
$S$-rings $\mathcal{K}$ if it is determined up to isomorphism in $\mathcal{K}$
only by the tensor of its structure constants. An abelian group is said to be
separable if every $S$-ring over this group is separable with respect to the
class of $S$-rings over abelian groups. Let $C_n$ be a cyclic group of order
$n$ and $G$ be a noncylic abelian $p$-group. From the previously obtained
results it follows that if $G$ is separable then $G$ is isomorphic to
$C_p\times C_{p^k}$ or $C_p\times C_p\times C_{p^k}$, where $p\in \{2,3\}$ and
$k\geq 1$. We prove that the groups $D=C_p\times C_{p^k}$ are separable
whenever $p\in \{2,3\}$. From this statement we deduce that a given Cayley
graph over $D$ and a given Cayley graph over an arbitrary abelian group one can
check whether these graphs are isomorphic in time $|D|^{O(1)}$.
