  In an effort to search for Ly$\alpha$ emission from circum- and intergalactic
gas on scales of hundreds of kpc around $z\sim3$ quasars, and thus characterise
the physical properties of the gas in emission, we have initiated an extensive
fast-survey with the Multi Unit Spectroscopic Explorer (MUSE): Quasar Snapshot
Observations with MUse: Search for Extended Ultraviolet eMission (QSO MUSEUM).
In this work, we report the discovery of an enormous Ly$\alpha$ nebula (ELAN)
around the quasar SDSS~J102009.99+104002.7 at $z=3.164$, which we followed-up
with deeper MUSE observations. This ELAN spans $\sim297$ projected kpc, has an
average Ly$\alpha$ surface brightness ${\rm SB}_{\rm Ly\alpha}\sim
6.04\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ (within the $2\sigma$
isophote), and is associated with an additional four, previously unknown
embedded sources: two Ly$\alpha$ emitters and two faint active galactic nuclei
(one Type-1 and one Type-2 quasar). By mapping at high significance the
line-of-sight velocity in the entirety of the observed structure, we unveiled a
large-scale coherent rotation-like pattern spanning $\sim300$ km s$^{-1}$ with
a velocity dispersion of $<270$ km s$^{-1}$, which we interpret as a signature
of the inspiraling accretion of substructures within the quasar's host halo.
Future multiwavelength data will complement our MUSE observations, and are
definitely needed to fully characterise such a complex system. None the less,
our observations reveal the potential of new sensitive integral-field
spectrographs to characterise the dynamical state of diffuse gas on large
scales in the young Universe, and thereby witness the assembly of galaxies.
