  {A cyclic subspace code is a union of the orbits of subspaces contained in
it. In a recent paper, Gluesing-Luerssen et al. (Des. Codes Cryptogr. 89,
447-470, 2021) showed that the study of the distance distribution of a single
orbit cyclic subspace code is equivalent to the study of its intersection
distribution. In this paper we have proved that in the orbit of a subspace $U$
of $\mathbb{F}_{q^n}$ that has the stabilizer $\mathbb{F}_{q^t}^*(t \neq n)$,
the number of codeword pairs $(U,\alpha U)$ such that $\dim(U\cap \alpha U)=i$
for any $i,~ 0\leq i < \dim(U)$, is a multiple of $q^t(q^t+1)$, if
$\frac{n}{t}$ is an odd number. In the case of even $\frac{n}{t}$, if $U$
contains $\frac{q^{2tm}-1}{q^{2t}-1}~ (m\geq 0)$ distinct cyclic shifts of
$\mathbb{F}_{q^{2t}}$, then the number of codeword pairs $(U,\alpha U)$ with
intersection dimension $2tm$ is equal to $q^t+rq^t(q^t+1)$, for some
non-negative integer $r$; and the number of codeword pairs $(U,\alpha U)$ with
intersection dimension $i,~(i\neq 2tm)$ is a multiple of $q^t(q^t+1)$. Some
examples have been given to illustrate the results presented in the paper.
