  We present the first direct search for lepton flavour violating muon decay
mediated by a new light particle X, $\mu^+ \to \mathrm{e}^+\mathrm{X},
\mathrm{X} \to \gamma\gamma$. This search uses a dataset resulting from
$7.5\times 10^{14}$ stopped muons collected by the MEG experiment at the Paul
Scherrer Institut in the period 2009--2013. No significant excess is found in
the mass region 20--45 MeV/c$^2$ for lifetimes below 40 ps, and we set the most
stringent branching ratio upper limits in the mass region of 20--40 MeV/c$^2$,
down to $\mathcal{O}(10^{-11})$ at 90\% confidence level.
