  We present a systematic analysis of homogeneous and isotropic cosmologies in
a particular Horndeski model with Galileon shift symmetry, containing also a
$\Lambda$-term and a matter. The model, sometimes called Fab Five, admits a
rich spectrum of solutions. Some of them describe the standard late time
cosmological dynamic dominated by the $\Lambda$-term and matter, while at the
early times the universe expands with a constant Hubble rate determined by the
value of the scalar kinetic coupling. For other solutions the $\Lambda$-term
and matter are screened at all times but there are nevertheless the early and
late accelerating phases. The model also admits bounces, as well as peculiar
solutions describing "the emergence of time". Most of these solutions contain
ghosts in the scalar and tensor sectors. However, a careful analysis reveals
three different branches of ghost-free solutions, all showing a late time
acceleration phase. We analyze the dynamical stability of these solutions and
find that all of them are stable in the future, since all their perturbations
stay bounded at late times. However, they all turn out to be unstable in the
past, as their perturbations grow violently when one approaches the initial
spacetime singularity. We therefore conclude that the model has no viable
solutions describing the whole of the cosmological history, although it may
describe the current acceleration phase. We also check that the flat space
solution is ghost-free in the model, but it may acquire ghost in more general
versions of the Horndeski theory.
