  Using the Very Large Telescope's Spectrograph for INtegral Field Observation
in the Near-Infrared (VLT/SINFONI), we have obtained repeated AO-assisted, NIR
spectroscopy of the six central luminous, Wolf-Rayet (WR) stars in the core of
the very young (~1 Myr), massive and dense cluster R136, in the Large
Magellanic Cloud (LMC). We also de-archived available images that were obtained
with the Hubble Space Telescope's Space Telescope Imaging Spectrograph
(HST/STIS), and extracted high-quality, differential photometry of our target
stars to check for any variability related to binary motion.
  Previous studies, relying on spatially unresolved, integrated, optical
spectroscopy, had reported that one of these stars was likely to be a 4.377-day
binary. Our study set out to identify the culprit and any other short-period
system among our targets. However, none displays significant photometric
variability, and only one star, BAT99-112 (R136c), located on the outer fringe
of R136, displays a marginal variability in its radial velocities; we
tentatively report an 8.2-day period. The binary status of BAT99-112 is
supported by the fact that it is one of the brightest X-ray sources among all
known WR stars in the LMC, consistent with it being a colliding-wind system.
Follow-up observations have been proposed to confirm the orbital period of this
potentially very massive system.
