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Recently, few cosmological models with additional non-Riemannian volumeform(s) have been proposed. In this article we use Supernovae type Iaexperimental data to test one of these models which provides a unifieddescription of both dark energy via dynamically generated cosmological constantand dark matter as a "dust" fluid due to a hidden nonlinear Noether symmetry.It turns out that the model allows various scenarios of the future Universeevolution and in the same time perfectly fits contemporary observational data.Further, we investigate the influence of an additional inflaton field with astep like potential. With its help we can reproduce the Universe inflationepoch, matter dominated epoch and present accelerating expansion in a seamlessway. Interesting feature is that inflaton undergoes a finite change during itsevolution. It can be speculated that the inflaton asymptotic value is connectedto the vacuum expectation value of the Higgs field.

Cosmological solutions from models with unified dark energy and dark matter and with inflaton field