  In this work, we consider a ``reverse-engineering'' approach to construct
confining potentials that support exact, constant density kovaton solutions to
the classical Gross-Pitaevskii equation (GPE) also known as the nonlinear
Schr\"odinger equation (NLSE). In the one-dimensional case, the exact solution
is the sum of stationary kink and anti-kink solutions, i.e. a kovaton, and in
the overlapping region, the density is constant. In higher dimensions, the
exact solutions are generalizations of this wave function. In the absence of
self-interactions, the confining potential is similar to a smoothed out finite
square well with minima also at the edges. When self-interactions are added, a
term proportional to $\pm g \psi^{\ast}\psi$ gets added to the confining
potential and $\pm g M$, where $M$ is the norm, gets added to the total energy.
In the realm of stability analysis, we find (linearly) stable solutions in the
case with repulsive self-interactions which also are stable to self-similar
deformations. For attractive interactions, however, the minima at the edges of
the potential get deeper and a barrier in the center forms as we increase the
norm. This leads to instabilities at a critical value of $M$ (related to the
number of particles in the BEC). Comparing the stability criteria from
Derrick's theorem and Bogoliubov-de Gennes analysis stability results, we find
that both predict stability for repulsive self-interactions and instability at
a critical mass $M$ for attractive interactions. However, the numerical
analysis gives a much lower critical mass. The numerical analysis shows further
that the initial instabilities violate the symmetry $x\rightarrow-x$ assumed by
Derrick's theorem.
