  We show that the dynamics of particles in a one-dimensional harmonic trap
with hard-core interactions can be solvable for certain arrangements of unequal
masses. For any number of particles, there exist two families of unequal mass
particles that have integrable dynamics, and there are additional exceptional
cases for three, four and five particles. The integrable mass families are
classified by Coxeter reflection groups and the corresponding solutions are
Bethe ansatz-like superpositions of hyperspherical harmonics in the relative
hyperangular coordinates that are then restricted to sectors of fixed particle
order. We also provide evidence for superintegrability of these Coxeter mass
families and conjecture maximal superintegrability.
