  The N$_2$ and CO-rich and water-depleted comet C/2016 R2 (Pan-STARRS)
(hereafter `C/2016 R2') is a unique comet for detailed spectroscopic analysis.
We aim to explore the associated photochemistry of parent species, which
produces different metastable states and forbidden emissions, in this cometary
coma of peculiar composition. We re-analyzed the high-resolution spectra of
comet C/2016 R2, which were obtained in February 2018, using the UVES
spectrograph of the European Southern Observatory (ESO) Very Large Telescope
(VLT). Various forbidden atomic emission lines of [CI], [NI], and [OI] were
observed in the optical spectrum of this comet when it was at 2.8 au from the
Sun. The observed forbidden emission intensity ratios are studied in the
framework of a couple-chemistry emission model. The model calculations show
that CO$_2$ is the major source of both atomic oxygen green and red-doublet
emissions in the coma of C/2016 R2 (while for most comets it is generally
H$_2$O), whereas, CO and N$_2$ govern the atomic carbon and nitrogen emissions,
respectively. Our modelled oxygen green to red-doublet and carbon to nitrogen
emission ratios are higher by a factor {of 3}, when compared to the
observations. These discrepancies can be due to uncertainties associated with
photon cross sections or unknown production/loss sources. Our modelled oxygen
green to red-doublet emission ratio is close to the observations, when we
consider an O$_2$ abundance with a production rate of 30\% relative to the CO
production rate. The collisional quenching is not a significant loss process
for N($^2$D) though its radiative lifetime is significant ($\sim$10 hrs).
Hence, the observed [NI] doublet-emission ratio ([NI] 5198/5200) of 1.22, which
is smaller than the terrestrial measurement by a factor {1.4}, is mainly due to
the characteristic radiative decay of N($^2$D).
