  In a recent letter, arXiv:1509.00021, it was shown that the lensing of light
around rotating boson stars and Kerr black holes with scalar hair can exhibit
chaotic patterns. Since no separation of variables is known (or expected) for
geodesic motion on these backgrounds, we examine the 2D effective potentials
for photon trajectories, to obtain a deeper understanding of this phenomenon.
We find that the emergence of stable light rings on the background spacetimes,
allows the formation of "pockets" in one of the effective potentials, for open
sets of impact parameters, leading to an effective trapping of some
trajectories, dubbed quasi-bound orbits. We conclude that pocket formation
induces chaotic scattering, although not all chaotic orbits are associated to
pockets. These and other features are illustrated in a gallery of examples,
obtained with a new ray-tracing code, PYHOLE, which includes tools for a
simple, simultaneous visualization of the effective potential together with the
spacetime trajectory, for any given point in a lensing image. An analysis of
photon orbits allows us to further establish a positive correlation between
photon orbits in chaotic regions and those with more than one turning point in
the radial direction; we recall that the latter is not possible around Kerr
black holes. Moreover, we observe that the existence of several light rings
around a horizon (several fundamental orbits, including a stable one), is a
central ingredient for the existence of multiple shadows of a single hairy
black hole. We also exhibit the lensing and shadows by Kerr black holes with
scalar hair, observed away from the equatorial plane, obtained with PYHOLE.
