  The data from the event horizon telescope have provided a novel view of the
vicinity of the horizon of a black hole (BH), by imaging the region around the
light-ring. They have also raised hopes for measuring in the near future,
features of the image (or the shadow) related to higher order effects of
photons traveling in these regions, such as the appearance of higher order
bright rings. While the prospect of measuring these fine features of Kerr BHs
is very interesting in itself, there are some even more intriguing prospects
for observing novel features of possible non-Kerr objects, in the case that the
subjects of our images are not the BH solutions of GR. In the hope of
sufficient resolution being available in the future, we explore in this work
the structure and properties of null geodesics around a Hartle-Thorne spacetime
that includes a deformation from the Kerr spacetime characterised by the
quadrupole deformation $\delta q$. These spacetimes have been found to exhibit
a bifurcation of the equatorial light-ring to two off-equatorial light-rings in
a range of $\delta q$s and spin parameters. In addition to this, there is a
range of parameters where both the equatorial and the off-equatorial
light-rings are present. This results in the formation of a pocket that can
trap photons. We investigate the properties of these trapped orbits and find
that chaotic behaviour emerges. Some of these chaotic orbits are additionally
found to be "sticky" and get trapped close to periodic orbits for long times.
We also explore how these novel features affect the shadow and find that the
off-equatorial light-rings produce distinctive features that deform its
circular shape, while the chaotic behaviour associated to the pocket creates
features with fractal structure. These results are shown to be quite general,
extending to higher order Hartle-Thorne spacetimes.
