  Considering the defects of the previous work for estimating the anomalous
production rates of $e^+e^-\to \Upsilon(1S)\pi^+\pi^-,\,\Upsilon(2S)\pi^+\pi^-$
near the peak of the $\Upsilon(5S)$ resonance at $\sqrt s=10.87$ GeV [K.F. Chen
{\it et al}. (Belle Collaboration), Phys. Rev. Lett. {\bf 100}, 112001 (2008)],
we suggest a new scenario where the contributions from the direct dipion
transition and the final state interactions interfere to result in not only the
anomalously large production rates, but also the lineshapes of the differential
widths consistent with the experimental measurement when assuming the reactions
are due to the dipion emission of $\Upsilon(5S)$. At the end, we raise a new
puzzle that the predicted differential width
$d\Gamma(\Upsilon(5S)\to\Upsilon(2S)\pi^+\pi^-)/d\cos\theta$ has a discrepant
trend from the data while other predictions are well in accord with the data.
It should be further clarified by more accurate measurements carried by future
experiments.
