  We characterise the equilibrium landscape, the entire manifold of local
equilibrium states, of an interacting integrable quantum model. Focusing on the
isotropic Heisenberg spin chain, we describe in full generality two
complementary frameworks for addressing equilibrium ensembles: the functional
integral Thermodynamic Bethe Ansatz approach, and the lattice regularisation
transfer matrix approach. We demonstrate the equivalence between the two, and
in doing so clarify several subtle features of generic equilibrium states. In
particular we explain the breakdown of the canonical Y-system, which reflects a
hidden structure in the parametrisation of equilibrium ensembles.
