  The recently proposed exact quantum solution for two
delta-function-interacting particles with a mass-ratio 3:1 in a hard-wall box
[Y. Liu, F. Qi, Y. Zhang and S. Chen, iScience 22, 181 (2019)] violates the
conventional necessary condition for a Bethe Ansatz integrability, the
condition being that the system must be reducible to a superposition of
semi-transparent mirrors that is invariant under all the reflections it
generates. In this article, we found a way to relax this condition: some of the
semi-transparent mirrors of a known self-invariant mirror superposition can be
replaced by the perfectly reflecting ones, thus breaking the self-invariance.
The proposed name for the method is Asymmetric Bethe Ansatz (Asymmetric BA). As
a worked example, we study in detail the bound states of the nominally
non-integrable system comprised of a bosonic dimer in a delta-well. Finally, we
show that the exact solution of the Liu-Qi-Zhang-Chen problem is a particular
instance of the the Asymmetric BA.
