Small Hamiltonian systems are connected in a chain the ends of which are coupled to unequal heat baths, forcing the system out of equilibrium. Energy exchange is of a form that leads to integrable dynamics. A proof of ergodicity of both equilibrium and nonequilibrium steady states will be presented. This is followed by numerical results which show that unlike certain mechanical systems with chaotic microdynamics, marginal distributions of NESS in these integrable chains are not Gibbsian, leading to problems in the definition of “local temperature.”