  We systematically study the most general Lorentz-violating graviton mass
invariant under three-dimensional Eucledian group using the explicitly
covariant language. We find that at general values of mass parameters the
massive graviton has six propagating degrees of freedom, and some of them are
ghosts or lead to rapid classical instabilities. However, there is a number of
different regions in the mass parameter space where massive gravity can be
described by a consistent low-energy effective theory with cutoff
$\sim\sqrt{mM_{Pl}}$ free of rapid instabilities and vDVZ discontinuity. Each
of these regions is characterized by certain fine-tuning relations between mass
parameters, generalizing the Fierz--Pauli condition. In some cases the required
fine-tunings are consequences of the existence of the subgroups of the
diffeomorphism group that are left unbroken by the graviton mass. We found two
new cases, when the resulting theories have a property of UV insensitivity,
i.e. remain well behaved after inclusion of arbitrary higher dimension
operators without assuming any fine-tunings among the coefficients of these
operators, besides those enforced by the symmetries. These theories can be
thought of as generalizations of the ghost condensate model with a smaller
residual symmetry group. We briefly discuss what kind of cosmology can one
expect in massive gravity and argue that the allowed values of the graviton
mass may be quite large, affecting growth of primordial perturbations,
structure formation and, perhaps, enhancing the backreaction of inhomogeneities
on the expansion rate of the Universe.
