  The three-particle quantization condition is partially diagonalized in the
center-of-mass frame by using cubic symmetry on the lattice. To this end,
instead of spherical harmonics, the kernel of the Bethe-Salpeter equation for
particle-dimer scattering is expanded in the basis functions of different
irreducible representations of the octahedral group. Such a projection is of
particular importance for the three-body problem in the finite volume due to
the occurrence of three-body singularities above breakup. Additionally, we
study the numerical solution and properties of such a projected quantization
condition in a simple model. It is shown that, for large volumes, these
solutions allow for an instructive interpretation of the energy eigenvalues in
terms of bound and scattering states.
