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Using data samples of $102\times10^6$ $\Upsilon(1S)$ and $158\times10^6$ $\Upsilon(2S)$ events collected with the Belle detector, a first experimental search has been made for double-charmonium production in the exclusive decays $\Upsilon(1S,2S)\rightarrow J/\psi(\psi')+X$, where $X=\eta_c$, $\chi_{cJ} (J=~0,~1,~2)$, $\eta_c(2S)$, $X(3940)$, and $X(4160)$. No significant signal is observed in the spectra of the mass …