  The spectral problem for an integrable system of particles satisfying the
fusion rules of $SU(3)_k$ is expressed in terms of exact inversion identities
satisfied by the commuting transfer matrices of the integrable fused
$A_2^{(1)}$ interaction round a face (IRF) model of Jimbo, Miwa and Okado. The
identities are proven using local properties of the Boltzmann weights, in
particular the Yang-Baxter equation and unitarity. They are closely related to
the consistency conditions for the construction of eigenvalues obtained in the
Separation of Variables approach to integrable vertex models.
