  Recently, few cosmological models with additional non-Riemannian volume
form(s) have been proposed. In this article we use Supernovae type Ia
experimental data to test one of these models which provides a unified
description of both dark energy via dynamically generated cosmological constant
and 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 Universe
evolution and in the same time perfectly fits contemporary observational data.
Further, we investigate the influence of an additional inflaton field with a
step like potential. With its help we can reproduce the Universe inflation
epoch, matter dominated epoch and present accelerating expansion in a seamless
way. Interesting feature is that inflaton undergoes a finite change during its
evolution. It can be speculated that the inflaton asymptotic value is connected
to the vacuum expectation value of the Higgs field.
