  We present a general parametrization for the leading order terms in a
momentum power expansion of a non-universal Lorentz-violating, but rotational
invariant, kinematics and its implications for two-body decay thresholds. The
considered framework includes not only modified dispersion relations for
particles, but also modified energy-momentum conservation laws, something which
goes beyond effective field theory. As a particular and relevant example,
bounds on the departures from special relativistic kinematics from the
non-observation of vacuum Cherenkov radiation are discussed and compared with
those obtained within the effective field theory scenario.
