  We generate inherent structures, local potential-energy minima, of the
"$k$-space overlap potential" in two-dimensional many-particle systems using a
cooling and quenching simulation technique. The ground states associated with
the $k$-space overlap potential are stealthy ({\it i.e.,} completely suppress
single scattering of radiation for a range of wavelengths) and hyperuniform
({\it i.e.,} infinite wavelength density fluctuations vanish). However, we show
via quantitative metrics that the inherent structures exhibit a range of
stealthiness and hyperuniformity depending on the fraction of degrees of
freedom that are constrained. Inherent structures in two dimensions typically
contain five-particle rings, wavy grain boundaries, and vacancy-interstitial
defects. The structural and thermodynamic properties of inherent structures are
relatively insensitive to the temperature from which they are sampled,
signifying that the energy landscape is relatively flat and devoid of deep
wells. Using the nudged-elastic-band algorithm, we construct paths from
ground-state configurations to inherent structures and identify the transition
points between them. In addition, we use point patterns generated from a random
sequential addition (RSA) of hard disks, which are nearly stealthy, and examine
the particle rearrangements necessary to make the configurations absolutely
stealthy. We introduce a configurational proximity metric to show that only
small local, but collective, particle rearrangements are needed to drive
initial RSA configurations to stealthy disordered ground states. These results
lead to a more complete understanding of the unusual behaviors exhibited by the
family of "collective-coordinate" potentials to which the $k$-space overlap
potential belongs.
