Context. The gamma-ray binary LS 5039 and the isolated pulsar PSR J1825-1446 were proposed to have been formed in the supernova remnant (SNR) G016.8-01.1. Aims. We aim to obtain the Galactic trajectory of LS 5039 and PSR J1825-1446 to find their origin in the Galaxy, and in particular to check their association with SNR G016.8-01.1 to restrict their age. Methods. By means of radio and optical observations we obtained the proper motion and the space velocity of the sources. Results. The proper motion of PSR J1825-1446 corresponds to a transverse space velocity of 690 km/s at a distance of 5 kpc. Its Galactic velocity at different distances is not compatible with the expected Galactic rotation. The velocity and characteristic age of PSR J1825-1446 make it incompatible with SNR G016.8-01.1. There are no clear OB associations or SNRs crossing the past trajectory of PSR J1825-1446. We estimate the age of the pulsar to be 80-245 kyr, which is compatible with its characteristic age. The proper motion of LS 5039 is 7.09 and -8.82 mas/yr in right ascension and declination, respectively. The association of LS 5039 with SNR G016.8-01.1 is unlikely, although we cannot to discard it. The system would have had to be formed in the association Ser OB2 (at 2.0 kpc) if the age of the system is 1.0-1.2 Myr, or in the association Sct OB3 (distance 1.5-2 kpc) for an age of 0.1-0.2 Myr. If the system were not formed close to Ser OB2, the pseudo-synchronization of the orbit would be unlikely. Conclusions. PSR J1825-1446 is a high-velocity isolated pulsar ejected from the Galaxy. The distance to LS 5039, which needs to be constrained by future astrometric missions such as Gaia, is a key parameter for restricting its origin and age.
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