  We search for di-muon decays of a low-mass Higgs boson ($A^0$) in the fully
reconstructed decay chain of $\Upsilon(2S,3S) \to \pipi\Upsilon(1S)$,
$\Upsilon(1S) \to \g A^0$, $A^0 \to \mumu$. The $A^0$ is predicted by several
extensions of the Standard Model (SM), including the Next-to-Minimal
Supersymmetric Standard Model (NMSSM). NMSSM introduces a \CP-odd light Higgs
boson whose mass could be less than 10 \gevcc. The data samples used in this
analysis contain $92.8 \times 10^6$ $\Upsilon(2S)$ and $116.8 \times 10^6$
$\Upsilon(3S)$ events collected by the \babar\ detector. The $\Upsilon(1S)$
sample is selected by tagging the pion pair in the $\Upsilon(2S, 3S) \to
\pi^+\pi^-\Upsilon(1S)$ transitions. We find no evidence for $A^0$ production
and set 90% confidence level (C.L.) upper limits on the product branching
fraction $\mathcal{B}(\Upsilon(1S) \to \g A^0) \times \mathcal{B}(A^0 \to
\mumu)$ in the range of $(0.28 - 9.7)\times 10^{-6}$ for $0.212 \le m_{A^0} \le
9.20$ \gevcc. We also combine our results with previous \babar\ results of
$\Upsilon(2S,3S) \to \g A^0$, $A^0 \to \mumu$ to set limits on the effective
coupling ($f_{\Upsilon}$) of the \b-quark to the $A^0$, $f_{\Upsilon}^2 \times
\mathcal{B}(A^0 \to \mumu)$, at the level of $(0.29 - 40)\times 10^{-6}$ for
$0.212 \le m_{A^0} \le 9.2$ GeV/$c^2$.
