  A search for the decays of the Higgs and $Z$ bosons to a $\phi$ meson and a
photon is performed with a $pp$ collision data sample corresponding to an
integrated luminosity of 2.7 fb$^{-1}$ collected at $\sqrt{s}=$13 TeV with the
ATLAS detector at the LHC. No significant excess of events is observed above
the background, and 95% confidence level upper limits on the branching
fractions of the Higgs and $Z$ boson decays to $\phi\,\gamma$ of
1.4$\times$10$^{-3}$ and 8.3$\times$10$^{-6}$, respectively, are obtained.
