We report on the experimental search for the bound state of an $eta$ meson and $^{3}hspace{-0.03cm}mbox{He}$ nucleus performed using the WASA-at-COSY detector setup. In order to search for the $eta$-mesic nucleus decay, the $pdrightarrow$ $^{3}hspace{-0.03cm}mbox{He} 2gamma$ and $pdrightarrow$ $^{3}hspace{-0.03cm}mbox{He} 6gamma$ channels have been analysed. These reactions manifest the direct decay of $eta$ meson bound in $^{3}hspace{-0.03cm}mbox{He}$ nucleus. This non-mesonic decay channel has been considered for the first time. When taking into account only statistical errors, the obtained excitation functions reveal a slight indication for a possible bound state signal corresponding to an $^3$He-$eta$ nucleus width $Gamma$ above 20 MeV and binding energy $B_s$ between 0 and 15 MeV. However, the determined cross sections are consistent with zero in the range of the systematic uncertainty. Therefore, as final result we estimate only the upper limit for the cross section of the $eta$-mesic $^{3}hspace{-0.03cm}mbox{He}$ nucleus formation followed by the $eta$ meson decay which varies between $2$ nb and $15$ nb depending on possible bound state parameters.
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