  The confirmation of the existence of GZK cut-off was tortuous, leading to
activities to explore new physics, such as the cosmic-ray new components,
unidentified cosmic-ray origins, unknown propagation mechanism, and the
modification of fundamental physics concepts like the tiny Lorentz invariance
violation (LV). The confirmation of the GZK cut-off provides an opportunity to
constrain the LV effect. We use a phenomenological framework to restudy the GZK
mechanism under the Planck scale deformation of the proton and pion dispersion
relations. Restudying the photon induced pion production of the proton
$\mathrm{p}+\gamma\to\mathrm{p}+\pi^0$, we predict abnormal threshold behaviors
of this reaction under different LV modifications. Therefore, we can study the
LV effects not only from the conventional GZK cut-off, but also from
potentially threshold anomalies of the pion production process. We divide the
LV parameter space into three regions, and analyze the constraints from current
observations in each region. The current observations have set strict
constraints on a certain LV region. However, for others LV regions, further
experimental observations and theoretical researches are still needed, and we
also find survival space for some theoretical explorations that permit specific
LV effects.
