  Here we consider isofrequency pairing of geodesic orbits that share the same
three orbital frequencies associated with $\Omega^{\hat{r}}$,
$\Omega^{\hat{\varphi}}$, and $\Omega^{\hat{\omega}}$ in a particular region of
parameter space around black string spacetime geometry. We study the effect
arising from the extra compact spatial dimension on the isofrequency pairing of
geodesic orbits and show that such orbits would occur and become in a
cylindrical manner in the allowed region as particles move along the black
string. We find that the effect due to the extra dimension leads to an increase
in the number of the isofrequency pairing of geodesic orbits. We also find that
isofrequency pairing of geodesic orbits in the region of parameter space can
not be realized beyond the critical value $J_{cr} \approx 0.096$ of the
conserved quantity of the motion due to the extra compact spatial dimension for
given parameters of the black string.
