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.