  We present the first strong-gravitational-lensing analysis of the galaxy
cluster RXJ0437.1+0043 (RXJ0437; z = 0.285). Newly obtained, deep MUSE
observations, Keck/MOSFIRE near-infrared spectroscopy, and Hubble Space
Telescope SNAPshot imaging reveal 13 multiply imaged background galaxies, three
of them (at z=1.98, 2.97, and 6.02, respectively) in hyperbolic umbilic (H-U)
lensing configurations. The H-U images are located only 20 -- 50 kpc from the
cluster centre, i.e., at distances well inside the Einstein radius where images
from other lens configurations are demagnified and often unobservable.
Extremely rare (only one H-U lens was known previously) these systems are able
to constrain the inner slope of the mass distribution -- and unlike radial
arcs, the presence of H-U configurations is not biased towards shallow cores.
The galaxies lensed by RXJ0437 are magnified by factors ranging from 30 to 300
and (in the case of H-U systems) stretched nearly isotropically. Taking
advantage of this extreme magnification, we demonstrate how the source galaxies
in H-U systems can be used to probe for small-scale ($\sim 10^{9} M_{\odot}$)
substructures, providing additional insight into the nature of dark matter.
