  Studying the flow of baryons into and out of galaxies is an important part of
understanding the evolution of galaxies over time. We present a detailed case
study of the environment around an intervening Ly $\alpha$ absorption line
system at $z_{\rm abs} = 0.633$, seen towards the quasar J0423$-$0130 ($z_{\rm
QSO} = 0.915$). We detect with ALMA the $^{12}$CO(2--1), $^{12}$CO(3--2) and
$1.2$~mm continuum emission from a galaxy at the redshift of the Ly $\alpha$
absorber at a projected distance of $135$ kpc. From the ALMA detections, we
infer ISM conditions similar to those in low redshift Luminous Infrared
Galaxies. DDT MUSE integral field unit observations reveal the optical
counterpart of the $^{12}$CO emission line source and three additional emission
line galaxies at the absorber redshift, which together form a galaxy group. The
$^{12}$CO emission line detections originate from the most massive galaxy in
this group. While we cannot exclude that we miss a fainter host, we reach a
dust-uncorrected star-formation rate (SFR) limit of > $0.3 \text{M}_{\odot}
\text{ yr}^{-1}$ within $100$ kpc from the sightline to the background quasar.
We measure the dust-corrected SFR (ranging from $3$ to $50$ M$_{\odot}$
yr$^{-1}$), the morpho-kinematics and the metallicities of the four group
galaxies to understand the relation between the group and the neutral gas
probed in absorption. We find that the Ly $\alpha$ absorber traces either an
outflow from the most massive galaxy or intra-group gas. This case study
illustrates the power of combining ALMA and MUSE to obtain a census of the cool
baryons in a bounded structure at intermediate redshift.
