  Context: Lyman $\alpha$ blobs (LABs) are large-scale radio-quiet Lyman
$\alpha$ (Ly$\alpha$) nebula at high-$z$ that occur predominantly in overdense
proto-cluster regions. Especially the prototypical SSA22a-LAB1 at $z=3.1$ has
become an observational reference for LABs across the electromagnetic spectrum.
  Aims: We want to understand the powering mechanisms that drive the LAB to
gain empirical insights into galaxy formation processes within a rare dense
environment at high-$z$.
  Methods: LAB 1 was observed for 17.5h with the VLT/MUSE integral-field
spectrograph. We produced optimally extracted narrow band images in Ly$\alpha$
$\lambda1216$ and HeII $\lambda1640$. By using a moment based analysis we
mapped the kinematics of the blob.
  Results: We detect Ly$\alpha$ emission to surface-brightness limits of
$10^{-19}$erg s$^{-1}$cm$^{-2}$arcsec$^{-2}$. At this depth we reveal a bridge
between LAB 1 and its northern neighbour LAB 8, as well as a shell-like
filament towards the south of LAB 1. We find a coherent large scale east-west
$\sim$1000 km s$^{-1}$ velocity gradient that is aligned perpendicular to the
major axis of the blob. We detect HeII emission in three distinct regions, but
we can only provide upper limits for CIV.
  Conclusions: Various gas excitation mechanisms are at play in LAB 1: Ionising
radiation and feedback effects dominate near the embedded galaxies, while
Ly$\alpha$ scattering is contributing at larger distances. However,
HeII/Ly$\alpha$ ratios combined with upper limits on CIV/Ly$\alpha$ can not
discriminate between AGN ionisation and feedback driven shocks. The alignment
of the angular momentum vector parallel to the morphological principal axis
appears odds with the predicted norm for high-mass halos, but likely reflects
that LAB\,1 resides at a node of multiple intersecting filaments of the cosmic
web.
  (Abridged)
