  A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of
$G$ is called normal if the right regular representation of $G$ is a normal
subgroup in the full automorphism group of $Cay(G,S)$, and is called a
CI-(di)graph if for any Cayley (di)graph $Cay(G,T)$, $Cay(G,S)\cong Cay(G,T)$
implies that there is $\sigma\in Aut(G)$ such that $S^\sigma=T$. A group $G$ is
called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley
digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all
Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively.
  DCI-groups and CI-groups have been widely studied in the literature. A lot of
efforts had been made to attach the classifications of cyclic DCI-groups and
CI-groups, motivated by the \'Ad\'am's conjecture, and finally Muzychuk
finished the classifications by proving that a cyclic group of order $n$ is a
DCI-group if and only if $n=mk$ where $m=1,2,4$ and $k$ is odd square-free, and
is a CI-group if and only if either $n=8,9,18$, or $n=mk$ where $m=1,2,4$ and
$k$ is odd square-free. In this paper we prove that a cyclic group of order $n$
is a NDCI-group if and only if $8\nmid n$, and is a NCI-group if and only if
either $n=8$ or $8\nmid n$.
