Bound states of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>Be</mml:mi> <mml:mprescripts/> <mml:mrow> <mml:mi>c</mml:mi> <mml:mover accent="true"> <mml:mi>c</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow

The possible bound state of the $_{c\overline{c}}^{9}\mathrm{Be}$, a charmonium-nucleus system, is investigated. The analysis is carried out within a three-cluster model, where its binary subsystems are represented as $c\overline{c}\text{+}\ensuremath{\alpha}$ and $\ensuremath{\alpha}+\ensuremath{\alpha}$. The hyperspherical harmonics method is employed to facilitate a convenient description of this three-cluster configuration. The calculations are done by employing the effective $c\overline{c}\text{\ensuremath{-}}\ensuremath{\alpha}$ potentials. These potentials were derived recently based on state-of-the-art lattice QCD calculations by HAL collaboration, which provided interactions for the spin-3/2 $J/\ensuremath{\psi}N$, spin-1/2 $J/\ensuremath{\psi}N$, spin-1/2 ${\ensuremath{\eta}}_{c}N$, and spin-averaged $J/\ensuremath{\psi}N$ interactions, all obtained with nearly physical pion masses. The Coulomb interaction was also incorporated into the current calculations. It is determined that, despite neither the $_{c\overline{c}}^{5}\mathrm{He}$ nor the $^{8}\mathrm{Be}$ binary subsystems being bound, a bound state of the $c\overline{c}\text{\ensuremath{-}}\ensuremath{\alpha}\ensuremath{\alpha}$ nuclear system could potentially exist. The maximum central binding energy is found to be approximately 1.71 MeV, based on the spin-1/2 $J/\ensuremath{\psi}N$ interaction, while a minimum value of about 0.56 MeV is obtained from calculations involving the spin-1/2 ${\ensuremath{\eta}}_{c}N$ interaction.