  The combined effects of the Lorentz-symmetry violating Chern-Simons and
Ricci-Cotton actions are investigated for the Einstein-Hilbert gravity in the
second order formalism modified by higher derivative terms, and their
consequences on the spectrum of excitations are analyzed. We follow the lines
of previous works and build up an orthonormal basis of operators that splits
the fundamental fields according to their individual degrees of freedom. With
this new basis, the attainment of the propagators is remarkably simplified and
the identification of the physical and unphysical modes gets a new insight. Our
conclusion is that the only tachyon- and ghost-free model is the
Einstein-Hilbert action added up by the Chern-Simons term with a time-like
vector of the type $v^{\mu} = (\mu,\vec{0})$. Spectral consistency imposes taht
the Ricci-Cotton term must be switched off. We then infer that gravity with
Lorentz-symmetry violation imposes a drastically different constraint on the
background if compared to usual gauge theories whenever conditions for
suppression of tachyons and ghosts are required.
