The middle-aged PSR J0357+3205 is a nearby , radio-quiet , bright \gamma -ray pulsar discovered by the Fermi mission .
Our previous Chandra observation revealed a huge , very peculiar structure of diffuse X-ray emission , originating at the pulsar position and extending for > 9 ^ { \prime } on the plane of the sky .
To better understand the nature of such a nebula , we have studied the proper motion of the parent pulsar .
We performed relative astrometry on Chandra images of the field spanning a time baseline of 2.2 yr , unveiling a significant angular displacement of the pulsar counterpart , corresponding to a proper motion of 0 \farcs 165 \pm 0 \farcs 030 yr ^ { -1 } .
At a distance of \sim 500 pc , the space velocity of the pulsar would be of \sim 390 km s ^ { -1 } assuming no inclination with respect to the plane of the sky .
The direction of the pulsar proper motion is perfectly aligned with the main axis of the X-ray nebula , pointing to a physical , yet elusive link between the nebula and the pulsar space velocity .
No optical emission in the H _ { \alpha } line is seen in a deep image collected at the Gemini telescope , which implies that the interstellar medium into which the pulsar is moving is fully ionized .