and $\vec p$ becomes curved. Therefore, when subsystems are composed to a whole, it is the pseudo-variables which must be additive in order to preserve the underlying Lorentz group structure. For this reason in the calculations of reaction thresholds or cross sections of particle interactions in the framework of DSR, conservation rules are formulated in terms of them [@JV; @LN]. This leads to a violation of ordinary energy/momentum conservation; particularly the total energy/momentum of a composite system never exceeds the invariant values, as long as the energy/momentum of its components are below them. (This is the so-called soccer-ball problem, see, e.g. [@Sta; @soc].)

Possible physical consequence are anomalies of reaction thresholds and an energy dependence of the speed of light in some versions of DSR. Even if the effect is tiny (of the order [*photon energy/Planck energy*]{}), it might become measurable when photons run over cosmic distances. A recently observed slight energy dependence of the time of arrival of photons from a $\gamma$ ray flare might be interpreted in this sense, if we knew the mechanism of emission, see [@magic].

More advanced versions of DSR are completed by modified space-time Lorentz transformations, associated to the transformations in momentum space in 