the characteristics (\[kinema-piN\]) do not necessary originate from $\eta$-mesic nuclei decays. They can also be produced by slow etas in a background nonresonance process shown in Fig. \[diagrams-piN\]b. The resonance and nonresonance processes correspond to a resonance (Breit-Wigner) and nonresonance part of the full propagator \[i.e. the Green function $G({\bm r}_1,
{\bm r}_2; E_\eta)$\] of the $\eta$-meson moving in the optical potential $U(r)$. Jointly, these parts generate a complicated spectrum of $E_\eta$ similar to that obtained in a toy model with a square-well potential [@lvov98; @sokol99]. Shown in Fig. \[spectral-function\] is the spectral function in that model, S(E\_) = ([r]{}\_1)  ([r]{}\_2)   |G([r]{}\_1, [r]{}\_2; E\_)|\^2 d[r]{}\_1 d[r]{}\_2, that characterizes near-threshold energy distribution of the propagated etas as well as the near-threshold energy dependence of the yield of $\pi N$ pairs produced by these $\eta$. Bound states of the $\eta$-meson give pronounced peaks in the yield of the $\pi N$ pairs at subthreshold energies $E_\eta$. Generally, observation of a relatively narrow resonance peak in the spectrum of $E_\eta$ in the region $E_\eta < m_\eta$ is mandatory for claiming an observation of $\eta$-mesic nuclei at all. We refer to recent works by Haider and Liu [@haider10a; @haider10b] where a deeper and more elaborated 