  A search for a new pseudoscalar $a$-boson produced in events with a top-quark
pair, where the $a$-boson decays into a pair of muons, is performed using
$\sqrt{s} = 13$ TeV $pp$ collision data collected with the ATLAS detector at
the LHC, corresponding to an integrated luminosity of $139\, \mathrm{fb}^{-1}$.
The search targets the final state where only one top quark decays to an
electron or muon, resulting in a signature with three leptons $e\mu\mu$ and
$\mu\mu\mu$. No significant excess of events above the Standard Model
expectation is observed and upper limits are set on two signal models: $pp
\rightarrow t\bar{t}a$ and $pp \rightarrow t\bar{t}$ with $t \rightarrow H^\pm
b$, $H^\pm \rightarrow W^\pm a$, where $a\rightarrow\mu\mu$, in the mass ranges
$15$ GeV $ < m_a < 72$ GeV and $120$ GeV $ \leq m_{H^{\pm}} \leq 160$ GeV.
