  A recently introduced one-dimensional two-particle Bose-Hubbard model with a
single impurity is studied on finite lattices. The model possesses a discrete
reflection symmetry and we demonstrate that all eigenstates odd under this
symmetry can be obtained with a generalized Bethe ansatz if periodic boundary
conditions are imposed. Furthermore, we provide numerical evidence that this
holds true for open boundary conditions as well. The model exhibits
backscattering at the impurity site -- which usually destroys integrability --
yet there exists an integrable subspace. We investigate the non-integrable even
sector numerically and find a class of states which have almost the Bethe
ansatz form. These weakly diffractive states correspond to a weak violation of
the non-local Yang-Baxter relation which is satisfied in the odd sector. We
bring up a method based on the Prony algorithm to check whether a numerically
obtained wave function is in the Bethe form or not, and if so, to extract
parameters from it. This technique is applicable to a wide variety of other
lattice models.
