  A detailed study of an inhomogeneous dust cosmology contained in a
$\gamma$-law family of perfect-fluid metrics recently presented by Mars and
Senovilla is performed. The metric is shown to be the most general orthogonally
transitive, Abelian, $G_2$ on $S_2$ solution admitting an additional homothety
such that the self-similar group $H_3$ is of Bianchi type VI and the fluid flow
is tangent to its orbits. The analogous cases with Bianchi types I, II, III, V,
VIII and IX are shown to be impossible thus making this metric privileged from
a mathematical viewpoint. The differential equations determining the metric are
partially integrated and the line-element is given up to a first order
differential equation of Abel type of first kind and two quadratures. The
solutions are qualitatively analyzed by investigating the corresponding
autonomous dynamical system. The spacetime is regular everywhere except for the
big bang and the metric is complete both into the future and in all spatial
directions. The energy-density is positive, bounded from above at any instant
of time and with an spatial profile (in the direction of inhomogeneity) which
is oscillating with a rapidly decreasing amplitude. The generic asymptotic
behaviour at spatial infinity is a homogeneous plane wave. Well-known dynamical
system results indicate that this metric is very likely to describe the
asymptotic behaviour in time of a much more general class of inhomogeneous
$G_2$ dust cosmologies.
