  Strongly lensed systems with peculiar configurations allow us to probe the
local properties of the deflecting lens mass while simultaneously testing
general profile assumptions. The quasar HE0230$-$2130 is lensed by two galaxies
at similar redshifts ($\Delta z \sim 0.003$) into four observed images. Using
modeled quasar positions from fitting the brightness of the quasar images in
ground-based imaging data from the Magellan telescope, we find that lens-mass
models where each of these two galaxies is parametrized with a singular
power-law (PL) profile predict five quasar images. To interpret the quad
configuration of the system, we tested 12 different profile assumptions with
the aim of obtaining lens-mass models that correctly predict only four observed
images. We tested the effects of adopting: cored profiles for the lensing
galaxies; external shear; and additional profiles to represent a dark matter
clump. We find that half of our model classes can produce the correct image
multiplicity. By comparing the Bayesian evidence of different model
parametrizations, we favor two model classes: (i) one that incorporates two
singular PL profiles for the lensing galaxies and a cored isothermal sphere in
the region of the previously predicted fifth image (rNIS profile), and (ii) one
with a bigger lensing galaxy parametrized by a singular PL profile and the
smaller galaxy by a cored PL profile with external shear. We estimated the mass
of the rNIS clump for each candidate model of our final Markov chain Monte
Carlo sample, and find that only 2\% are in the range of $10^6 M_{\odot} \leq
M_{\rm rNIS}\leq 10^9 M_{\odot}$, which is the predicted mass range of dark
matter subhalos in cold dark matter simulations, or the mass of
dark-matter-dominated and low-surface-brightness galaxies. We therefore favor
the models with a cored mass distribution for the lens galaxy close to the
predicted fifth image.
