  Context. Protoplanetary disks around young stars often contain substructures
like rings, gaps, and spirals that could be caused by interactions between the
disk and forming planets. Aims. We aim to study the young (1-3 Myr) star DR Tau
in the near-infrared and characterize its disk, which was previously resolved
through sub-millimeter interferometry with ALMA, and to search for possible
sub-stellar companions embedded into it. Methods. We observed DR Tau with
VLT/SPHERE both in polarized light (H broad band) and total intensity (in Y, J,
H, and K spectral bands). We also performed L' band observations with
LBTI/LMIRCam on the Large Binocular Telescope (LBT). Results. We found two
previously undetected spirals extending north-east and south of the star,
respectively. We further detected an arc-like structure north of the star.
Finally a bright, compact and elongated structure was detected at separation of
303 +/- 10 mas and position angle 21.2 +/- 3.7 degrees, just at the root of the
north-east spiral arm. Since this feature is visible both in polarized light
and in total intensity and has a flat spectrum it is likely caused by stellar
light scattered by dust. Conclusions. The two spiral arms are at different
separation from the star, have very different pitch angles, and are separated
by an apparent discontinuity, suggesting they might have a different origin.
The very open southern spiral arm might be caused by infalling material from
late encounters with cloudlets into the formation environment of the star
itself. The compact feature could be caused by interaction with a planet in
formation still embedded in its dust envelope and it could be responsible for
launching the north-east spiral. We estimate a mass of the putative embedded
object of the order of few M_Jup .
