  Context. Deep spectroscopic surveys with the Atacama Large
Millimeter/submillimeter Array (ALMA) have revealed that some of the brightest
infrared sources in the sky correspond to concentrations of dusty star-forming
galaxies (DSFG) at high redshift. Among these, the SPT2349-56 protocluster
system at z = 4.304 is amongst the most extreme examples due to its high source
density and integrated star formation rate. Aims. We conducted a deep
Lyman-$\alpha$ line emission survey around SPT2349-56 using the Multi-Unit
Spectroscopic Explorer (MUSE) at Very Large Telescope (VLT) in order to
characterize this uniquely dense environment. Methods. Taking advantage of the
deep three-dimensional nature of this survey, we performed a sensitive search
for Lyman-$\alpha$ emitters (LAEs) toward the core and northern extension of
the protocluster, which correspond to the brightest infrared regions in this
field. Using a smoothed narrowband image extracted from the MUSE datacube
around the protocluster redshift, we searched for possible extended structures.
Results. We identify only three LAEs at z = 4.3 in this field, in concordance
with expectations for blank-fields, and an extended Lyman-$\alpha$ structure
spatially associated with core of the protocluster. All the
previously-identified DSFGs in this field are undetected in Lyman-$\alpha$
emission, consistent with the conspicuous dust obscuration in these systems. We
find an extended Lyman-$\alpha$ structure, about $60 \times 60$ kpc$^{2}$ in
size, and located 56 kpc west of the protocluster core. Three DSFGs coincide
spatially with the location of this structure. We conclude that either the
three co-spatial DSFGs or the protocluster core itself are feeding ionizing
photons to the Lyman-$\alpha$ structure.
