  We show that a system consisting of two interacting particles with mass ratio
$3$ or $1/3$ in a hard-wall box can be exactly solved by using Bethe-type
ansatz. The ansatz is based on a finite superposition of plane waves associated
with a dihedral group $D_{6}$, which enforces the momentums after a series of
scattering and reflection processes to fulfill the $D_{6}$ symmetry. Starting
from a two-body elastic collision model in a hard-wall box, we demonstrate how
a finite momentum distribution is related to the $D_{2n}$ symmetry for
permitted mass ratios. For a quantum system with mass ratio $3$, we obtain
exact eigenenergies and eigenstates by solving Bethe-type-ansatz equations for
arbitrary interaction strength. A many-body excited state of the system is
found to be independent of the interaction strength, i.e. the wave function
looks exactly the same for non-interacting two particles or in the hard-core
limit.
