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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-typeansatz. The ansatz is based on a finite superposition of plane waves associatedwith a dihedral group $D_{6}$, which enforces the momentums after a series ofscattering and reflection processes to fulfill the $D_{6}$ symmetry. Startingfrom a two-body elastic collision model in a hard-wall box, we demonstrate howa finite momentum distribution is related to the $D_{2n}$ symmetry forpermitted mass ratios. For a quantum system with mass ratio $3$, we obtainexact eigenenergies and eigenstates by solving Bethe-type-ansatz equations forarbitrary interaction strength. A many-body excited state of the system isfound to be independent of the interaction strength, i.e. the wave functionlooks exactly the same for non-interacting two particles or in the hard-corelimit.

Mass-imbalanced atoms in a hard-wall trap: an exactly solvable model with emergent $D_{6}$ symmetry