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

The positronium, a bound state of electron and positron, is a unique system to perform highly precise tests, due to no hadronic background and precise Quantum Electrodynamics (QED) predictions. Being a system of lepton and antilepton, its properties are precisely described by QED in the Standard Model (SM). The final events topology can be simulated using Monte Carlo techniques. The J-PET detector is a multi-purpose, large acceptance system that is very well-suitable to the studies of positronium decay due to its excellent angular (1) and timing resolutions. We present preliminary results on the feasibility of searching for Dark Matter (DM) candidates in the decay o-Ps invisible with the J-PET, which is well-suited for the detection of positronium-decay products. Toy Monte Carlo simulations have been prepared to incorporate DM decay models to the o-Ps decay expectations in order to assess the detector capabilities to search for such an elusive component of our Universe.