  The effect of self-gravity of a disk matter is evaluated by the simplest
modes of oscillation frequencies for perturbed circular geodesics. It is
plotted the radial profiles of free oscillations of an equatorial circular
geodesic perturbed within the orbital plane or in the vertical direction. The
calculation is carried out to geodesics of an axisymmetric $n$-dimensional
spacetime. The profiles are computed by examples of disks embeded in
five-dimensional or six-dimensional spacetime, where it is studied the motion
of free test particles for three axisymmetric cases: (i) the Newtonian limit of
a general proposed $5D$ and $6D$ axisymmetric spacetime; (ii) a simple
Randall-Sundrum $5D$ spacetime; (iii) general $5D$ and $6D$ Randall-Sundrum
spacetime. The equation of motion of such particles is derived and the
stability study is computed for both horizontal and vertical directions, to see
how extra dimensions could affect the system. In particular, we investigate a
disk constructed from Schwarzschild and Chazy-Curzon solutions with a simple
extension for extra dimensions in the case (i), and by solving vacuum Einstein
field equations for a kind of Randall-Sundrum-Weyl metric in cases (ii) and
(iii). We find that it is possible to compute a range of possible solutions
where such perturbed geodesics are stable. Basicaly, the stable solutions
appear, for the radial direction, in special cases when the system has $5D$ and
in all cases when the system has $6D$; and, for the axial direction, in all
cases when the system has both $5D$ or $6D$.
