Search for $\overline{K}NN$ bound states with the FINUDA spectrometer

The existence of a deeply-bound kaonic state, which includes an antikaon inside the nucleus, is theoretically predicted by Akaishi and Yamazaki, according to their $\overline{K}N$ potential. In their model, the $\Lambda$(1405) is regarded as a $\overline{K}N$ bound state. We have searched for the lightest kaon-bound system ($\overline{K}NN$) with the FINUDA spectrometer, which is installed at the $e^+e^-$ collider DA$\Phi$NE. The $\phi$-meson, abundantly produced by DA$\Phi$NE, decays into $K^+K^-$, with the kinetic energy of $K^\pm$ $\sim 16\,\mathrm{MeV}$. This slow $K^-$ stops inside a very thin nucleus target and interacts with a nucleus. We installed five kinds of targets ($^6$Li, $^7$Li, $^{12}$C, $^{27}$Al and $^{51}$V) surrounding the beam pipe in the first run (2003--2004). In this talk, I will discuss non-mesonic decay modes of a $\overline{K}NN$ bound system, clearly observed in the back-to-back angular correlation between a hyperon and a nucleon emitted from the $K^-$ reaction vertex.