  Entanglement detection in high dimensional systems is a NP-hard problem since
it is lacking an efficient way. Given a bipartite quantum state of interest
free entanglement can be detected efficiently by the PPT-criterion
(Peres-Horodecki criterion), in contrast to detecting bound entanglement, i.e.
a curious form of entanglement that can also not be distilled into maximally
(free) entangled states. Only a few bound entangled states have been found,
typically by constructing dedicated entanglement witnesses, so naturally the
question arises how large is the volume of those states. We define a large
family of magically symmetric states of bipartite qutrits for which we find
$82\%$ to be free entangled, $2\%$ to be certainly separable and as much as
$10\%$ to be bound entangled, which shows that this kind of entanglement is not
rare. Via various machine learning algorithms we can confirm that the remaining
$6\%$ of states are more likely to belonging to the set of separable states
than bound entangled states. Most important we find via dimension reduction
algorithms that there is a strong $2$-dimensional (linear) sub-structure in the
set of bound entangled states. This revealed structure opens a novel path to
find and characterize bound entanglement towards solving the long-standing
problem of what the existence of bound entanglement is implying.
