  Testing Lorentz invariance is essential as it is one of the pillars of modern
physics. Moreover, its violation is foreseen in several popular Quantum Gravity
models. Several authors study the effects of Lorentz invariance violation (LIV)
in the propagation of ultra-high energy cosmic rays. These particles are the
most energetic events ever detected and therefore represent a promising
framework to test LIV. In this work we present an analytic calculation of the
in-elasticity for any $a+b \rightarrow c+d$ interaction using first order
perturbation in the dispersion relation that violates Lorentz invariance. The
inelasticity can be calculated by solving a third-order polynomial equation
containing: a) the kinematics of the interaction, b) the LIV term for each
particle and c) the geometry of the interaction. We use the inelasticity we
calculate to investigate the proton propagation in the intergalactic media. The
photopion production of the proton interaction with the CMB is taken into
account using the inelasticity and the attenuation length in different LIV
scenarios. We show how the allowed phase space for the photopion production
changes when LIV is considered for the interaction. The calculations presented
here are going to be extended in order to calculated the modified ultra-high
energy cosmic rays spectrum and compare it to the data.
