Presented at the Symposium on New Trends in Nuclear and Medical Physics, Kraków, Poland, 18-20 October, 2023.

The positronium system, a bound state of an electron and a positron, is suitable for testing the predictions of quantum electrodynamics (QED) as well as symmetry invariance. The Ps triple state, the ortho-Positronium (o-Ps), which mainly decays to three photons, is further studied to search for decays into 4γ and 5γ, the former C-violating decay and the latter never observed. The J-PET is a multi-purpose detector optimized for the detection of photons from positron–electron annihilation and can be used in a broad scope of interdisciplinary investigation. The large acceptance and high angular resolution of the J-PET detector will push the present limits in these forbidden and rare decays. The aim is to reach a sensitivity below O(10) while reducing the uncertainties, thus increasing the sensitivity.