The five-moment closure in Levermore's moment hierarchy is considered as a model problem for an extended Euler system which involves velocity moments up to order four. It is obtained by taking moments of the one- dimensional Boltzmann equation under the assumption that the velocity distribution is a maximum-entropy function. The moment vectors for which a maximum-entropy function exist consequently make up the domain of definition of the system. The aim of this talk is to give a complete characterization of the structure of the domain of definition and the connected maximum-entropy problem. Properties of the flux function of the five moment system are also studied and numerical difficulties connected with the evaluation of the flux are addressed.