Pionium as a source of false events in the $K\to\pi\nu\bar\nu$ decays

We suggest that the decay modes of kaons with a pion and a pionium ($\pi^+\pi^-$ atom) in the final state can constitute a not yet considered background to the very rare decay $K\to\pi\nu\bar\nu$. In fact, a part of pioniums may escape the decay region before decaying into two $\pi^0$s (or to $\pi^0\pi^0\gamma$ in the case of excited pionium). To illustrate the importance of this background, we show that it may even explain, under some assumptions, the unexpected $K_L$ decay events that appeared in the KOTO experiment.