  The Einstein equations for one of the hypersurface-homogeneous rotating dust
models are investigated. It is a Bianchi type V model in which one of the
Killing fields is spanned on velocity and rotation (case 1.2.2.2 in the
classification scheme of the earlier papers). A first integral of the field
equations is found, and with a special value of this integral coordinate
transformations are used to eliminate two components of the metric. The k = -1
Friedmann model is shown to be contained among the solutions in the limit of
zero rotation. The field equations for the simplified metric are reduced to 3
second-order ordinary differential equations that determine 3 metric components
plus a first integral that algebraically determines the fourth component. First
derivatives of the metric components are subject to a constraint (a
second-degree polynomial with coefficients depending on the functions). It is
shown that the set does not follow from a Lagrangian of the Hilbert type. The
group of Lie point-symmetries of the set is found, it is two-dimensional
noncommutative. Finally, a method of searching for first integrals (for sets of
differential equations) that are polynomials of degree 1 or 2 in the first
derivatives is applied. No such first integrals exist. The method is used to
find a constraint (of degree 1 in first derivatives) that could be imposed on
the metric, but it leads to a vacuum solution, and so is of no interest for
cosmology.
