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The radio-quiet quasar Q2059-360 at redshift $z=3.08$ is known to be close toa small Lyman $\alpha$ blob (LAB) and to be absorbed by a proximate dampedLy$\alpha$ (PDLA) system.  Here, we present the Multi Unit Spectroscopic Explorer (MUSE) integral fieldspectroscopy follow-up of this quasi-stellar object (QSO). Our primary goal isto characterize this LAB in detail by mapping it both spatially and spectrallyusing the Ly$\alpha$ line, and by looking for high-ionization lines toconstrain the emission mechanism.  Combining the high sensitivity of the MUSE integral field spectrographmounted on the Yepun telescope at ESO-VLT with the natural coronagraph providedby the PDLA, we map the LAB down to the QSO position, after robust subtractionof QSO light in the spectral domain.  In addition to confirming earlier results for the small bright component ofthe LAB, we unveil a faint filamentary emission protruding to the south overabout 80 pkpc (physical kpc); this results in a total size of about 120 pkpc.We derive the velocity field of the LAB (assuming no transfer effects) and mapthe Ly$\alpha$ line width. Upper limits are set to the flux of the N V $\lambda1238-1242$, C IV $\lambda 1548-1551$, He II $\lambda 1640$, and C III] $\lambda1548-1551$ lines. We have discovered two probable Ly$\alpha$ emitters at thesame redshift as the LAB and at projected distances of 265 kpc and 207 kpc fromthe QSO; their Ly$\alpha$ luminosities might well be enhanced by the QSOradiation. We also find an emission line galaxy at $z=0.33$ near the line ofsight to the QSO.  This LAB shares the same general characteristics as the 17 others surroundingradio-quiet QSOs presented previously. However, there are indications that itmay be centered on the PDLA galaxy rather than on the QSO.

MUSE-inspired view of the quasar Q2059-360, its Lyman alpha blob, and its neighborhood