We search for the $J^{PC}=0^{--}$ and $1^{+-}$ light tetraquark states with masses up to 2.46~GeV/$c^2$ in $Upsilon(1S)$ and $Upsilon(2S)$ decays with data samples of $(102pm 2)$ million and $(158pm 4)$ million events, respectively, collected with the Belle detector. No significant signals are observed in any of the studied production modes, and 90% credibility level (C.L.) upper limits on their branching fractions in $Upsilon(1S)$ and $Upsilon(2S)$ decays are obtained. The inclusive branching fractions of the $Upsilon(1S)$ and $Upsilon(2S)$ decays into final states with $f_1(1285)$ are measured to be ${cal B}(Upsilon(1S)to f_1(1285)+anything)=(46pm28({rm stat.})pm13({rm syst.}))times 10^{-4}$ and ${cal B}(Upsilon(2S)to f_1(1285)+anything)=(22pm15({rm stat.})pm6.3({rm syst.}))times 10^{-4}$. The measured $chi_{b2} to J/psi + anything$ branching fraction is measured to be $(1.50pm0.34({rm stat.})pm0.22({rm syst.}))times 10^{-3}$, and 90% C.L. upper limits for the $chi_{b0,b1} to J/psi + anything$ branching fractions are found to be $2.3times 10^{-3}$ and $1.1times 10^{-3}$, respectively. For ${cal B}(chi_{b1} to omega + anything)$, the branching fraction is measured to be $(4.9pm1.3({rm stat.})pm0.6({rm syst.}))times 10^{-2}$. %($<3.68times 10^{-2}$ at 90% C.L.). All results reported here are the first measurements for these modes.
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