  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 $(102\pm 2)$ million and $(158\pm 4)$ 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 ${\cal B}(\Upsilon(1S)\to f_1(1285)+anything)=(46\pm28({\rm
stat.})\pm13({\rm syst.}))\times 10^{-4}$ and ${\cal B}(\Upsilon(2S)\to
f_1(1285)+anything)=(22\pm15({\rm stat.})\pm6.3({\rm syst.}))\times 10^{-4}$.
The measured $\chi_{b2} \to J/\psi + anything$ branching fraction is measured
to be $(1.50\pm0.34({\rm stat.})\pm0.22({\rm syst.}))\times 10^{-3}$, and 90\%
C.L. upper limits for the $\chi_{b0,b1} \to J/\psi + anything$ branching
fractions are found to be $2.3\times 10^{-3}$ and $1.1\times 10^{-3}$,
respectively. For ${\cal B}(\chi_{b1} \to \omega + anything)$, the branching
fraction is measured to be $(4.9\pm1.3({\rm stat.})\pm0.6({\rm syst.}))\times
10^{-2}$. %($<3.68\times 10^{-2}$ at 90\% C.L.). All results reported here are
the first measurements for these modes.
