A search for decays of the Higgs boson into a pair of new spin-zero particles, $H \to aa$, where the $a$-bosons decay into a $b$-quark pair and muon pair, is presented. The search uses 36.1 fb$^{-1}$ of proton-proton collision data at $\sqrt{s}=13$ TeV recorded by the ATLAS experiment at the LHC in 2015 and 2016. No significant deviation from the Standard Model prediction is observed. Upper limits at 95% confidence level are placed on the branching ratio $(\sigma_H / \sigma_{SM}) \times B(H\to aa\to bb\mu\mu)$, ranging from $2\times 10^{-4}$ to $10^{-3}$ in the $a$-boson mass range of $20-60$ GeV. Model-independent limits are set on the visible production cross-section times the branching ratio to the $bb\mu\mu$ final state for new physics, $\sigma_{vis}(X) \times B(X\to bb\mu\mu)$, ranging from 0.1 fb to 0.73 fb for $m_{\mu\mu}$ between 18 and 62 GeV.

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