  Observing dynamical interactions between planets and disks is key to
understanding their formation and evolution. Two protoplanets have recently
been discovered within PDS 70's protoplanetary disk, along with an arm-like
structure towards the north-west of the star. Our aim is to constrain the
morphology and origin of this arm-like structure, and to assess whether it
could trace a spiral density wave caused by the dynamical interaction between
the planet PDS 70c and the disk. We analyze polarized and angular differential
imaging (PDI and ADI) data taken with VLT/SPHERE, spanning six years of
observations. PDI data sets are reduced using the IRDAP polarimetric data
reduction pipeline, while ADI data sets are processed using MUSTARD, a novel
inverse problem algorithm to tackle the geometrical biases spoiling the images
previously used for the analysis of this disk. We confirm the presence of the
arm-like structure in all PDI and ADI datasets. We do not observe a south-east
symmetric arm with respect to the disk minor axis, which seems to disfavor the
previous hypothesis that the arm is the footprint of a double-ring structure.
If the structure traces a spiral density wave following the motion of PDS 70c,
we would expect
$11\overset{\circ}{.}28^{+2\overset{\circ}{.}20}_{-0\overset{\circ}{.}86}$
rotation for the spiral in six years. However, we do not measure any
significant movement of the structure. If the arm-like structure is a
planet-driven spiral arm, the observed lack of rotation would suggest that the
assumption of rigid-body rotation may be inappropriate for spirals induced by
planets. We suggest that the arm-like structure may rather trace a vortex
appearing as a one-armed spiral in scattered light due to projection effects.
The vortex hypothesis accounts for both the lack of observed rotation and the
presence of a nearby sub-mm continuum asymmetry detected with ALMA.
