a\_[N]{} = 0.28 + i 0.19 fm. \[a-etaN-BL\] The positive value of ${\rm Re\,}a_{\eta N}$ means an effective attraction between $\eta$ and $N$, so that one can expect that several nucleons could jointly bind $\eta$ to a nuclear orbit. The first-order static-limit on-shell optical potential of $\eta$ in the nuclear matter at zero energy $E_\eta^{\rm kin}=0$ is equal to U(r) = -2 a\_[N]{}(r) ( + ), what gives \[together with Eq. (\[a-etaN-BL\])\] $U = -34 -i\;23$ MeV at normal nuclear matter density $\rho=\rho_0 = 0.17~\rm fm^{-3}$. The imaginary part of the potential describes a local absorption rate $\Gamma = -2\,{\rm Im}\,U$ of $\eta$ in the nuclear substance.

With the above strength of the $\eta A$ potential, $\eta$-mesic nuclei $_\eta A$ are expected to exist for all $A\ge 10$ [@haider02; @haider09]. Actually, due to a sharp (cusp) energy dependence of the $\eta N$ scattering amplitude near threshold, Fermi motion of nucleons and $\eta$ reduces the optical potential \[especially its imaginary part\], and this makes $\eta$-mesic nuclei to exist only for $A\ge 12$. For binding energies and widths of the lightest $\eta$-mesic nuclei Haider and Liu predicted [@haider02; @haider09] E\_= -1.19 [MeV]{}, && \_=   7.34 [MeV for\^[12]{}\_[ ]{}C]{}, E\_= -3.45 [MeV]{}, && 