A cosmological model with van der Waals gas and dust has been studied in the context of a three-component autonomous non-linear dynamical system involving the time evolution of the particle number density, the Hubble parameter and the temperature. Due to the presence of a symmetry of the model, the temperature evolution law is determined (in terms of the particle number density) and with this the dynamical system reduces to a two-component one which is fully integrable. The globally conserved Hamiltonian is identified and, in addition to it, some special (second) integrals, defined and conserved on a lower-dimensional manifold, are found. The parameter choices and their implication for the global dynamics in terms of cosmological relevance are comprehensively studied and the physically meaningful parameter values are identified.

25 pages, 21 figures, 1 table