  We study Galileon theories that emerge in ghost-free massive gravity. In
particular, we focus on a sub-class of these theories where the Galileons can
be completely decoupled from the tensor Lagrangian. These Galileons differ from
generic ones -- they have interrelated coefficients of the cubic and quartic
terms, and most importantly, a non-standard coupling to external
stress-tensors, governed by the same coefficient. We show that this theory has
no static stable spherically symmetric solutions that would interpolate from
the Vainshtein region to flat space; these two regions cannot be smoothly
matched for the sign of the coefficient for which fluctuations are stable.
Instead, for this sign choice, a solution in the Vainshtein domain is matched
onto a cosmological background. Small fluctuations above this solution are
stable, and sub-luminal. We discuss observational constraints on this theory,
within the quantum effective Lagrangian approach, and argue that having a
graviton mass of the order of the present-day Hubble parameter, is consistent
with the data. Last but not least, we also present a general class of
cosmological solutions in this theory, some of which exhibit the de-mixing
phenomenon, previously found for the self-accelerated solution.
