Since the middle of last century, researchers have observed EAS events with more than one pulse from the passage of particles in each detector. These events were called events with a delayed particle at the time, now they are also called multi-modal events. The first registration of such events was described by J. Jelley and W. Whitehouse [1] in 1953. Later, EAS exhibiting the unusual time structures were studied by several independent experiments, see e.g. [2]. EAS events with delayed particles were studied by British and US research groups in the 1960s-1980s, since the 1970s, delayed particles in EAS have been studied in Japan and also carried out at Moscow State University (MSU) and at LPI (TSHASS) and other groups. The conclusions that these articles give do not match and point to the lack of understanding of the nature of events with delayed particles.The Horizon-T experiment in Tien Shan is based on the idea of measuring the time at which EAS disc passes the observation level with nanosecond accuracy. The detector system [3] consists of ten charged particles registration points located at distances of up to several hundred meters from each other. The points are equipped with detectors based on registration of Cherenkov radiation in glass and registration of scintillation light in polystyrene. The detectors register the arrival times of charged particles at the observation level with a resolution of ∼2 ns, as required to study the spatial and temporal characteristics of the EAS and the structure of the multi-modal events specifically. Over the period of the Horizon-T data taking since 2017 to present, a large number of multi-modal events were detected. The data has presented numerous challenges that show the direction towards the further development of the detector system and of the analysis methods and techniques that could be applied to these multi-modal events [4].These recently observed Multi-Modal Cosmic Rays Events (MME) containing multiple peaks separated by tens to hundreds of ns are one of the most puzzling phenomena in high energy cosmic rays. These unusual cosmic ray events offer a potential new insight into the ultra-high energy astrophysics, the physics of fundamental particle interactions and cosmology. Surely, further experimental studies and independent verification of these events both at lower and high altitudes are needed to better understand the MME phenomenology. Theoretical insights are very much needed as well to guide future studies. It was recently proposed in [5] that these MMEs might be result of the dark matter annihilation events within the so-called axion quark nugget (AQN) dark matter model, which was originally invented for completely different purpose to explain the observed similarity between the dark and the visible components in the Universe, i.e. Ω DM ≈ Ω visible , without any fitting parameters.Earlier, in the HADRON experiment on Tien Shan, the possible presence of a non-nuclear component in primary cosmic rays was established and it was assumed that this component coul...