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PSR J1357$-$6429 is a young and energetic radio pulsar detected in X-rays and $gamma$-rays. It powers a compact pulsar wind nebula with a jet visible in X-rays and a large scale plerion detected in X-ray and TeV ranges. Previous multiwavelength studi es suggested that the pulsar has a significant proper motion of about 180 mas yr$^{-1}$ implying an extremely high transverse velocity of about 2000 km s$^{-1}$. In order to verify that, we performed radio-interferometric observations of PSR J1357$-$6429 with the the Australia Telescope Compact Array (ATCA) in the 2.1 GHz band. We detected the pulsar with a mean flux density of $212pm5$ $mu$Jy and obtained the most accurate pulsar position, RA = 13:57:02.525(14) and Dec = $-$64:29:29.89(15). Using the new and archival ATCA data, we did not find any proper motion and estimated its 90 per cent upper limit $mu < 106$ mas yr$^{-1}$. The pulsar shows a highly polarised single pulse, as it was earlier observed at 1.4 GHz. Spectral analysis revealed a shallow spectral index $alpha_{ u}$ = $0.5 pm 0.1$. Based on our new radio position of the pulsar, we disclaim its optical counterpart candidate reported before.
PSR J2021+3651 is a 17 kyr old rotation powered pulsar detected in the radio, X-rays, and $gamma$-rays. It powers a torus-like pulsar wind nebula with jets, dubbed the Dragonfly, which is very similar to that of the Vela pulsar. The Dragonfly is like ly associated with the extended TeV source VER J2019+368 and extended radio emission. We conducted first deep optical observations with the GTC in the Sloan $r$ band to search for optical counterparts of the pulsar and its nebula. No counterparts were detected down to $rgtrsim27.2$ and $gtrsim24.8$ for the point-like pulsar and the compact X-ray nebula, respectively. We also reanalyzed Chandra archival X-ray data taking into account an interstellar extinction--distance relation, constructed by us for the Dragonfly line of sight using the red-clump stars as standard candles. This allowed us to constrain the distance to the pulsar, $D=1.8^{+1.7}_{-1.4}$ kpc at 90% confidence. It is much smaller than the dispersion measure distance of $sim$12 kpc but compatible with a $gamma$-ray pseudo-distance of 1 kpc. Based on that and the optical upper limits, we conclude that PSR J2021+3651, similar to the Vela pulsar, is a very inefficient nonthermal emitter in the optical and X-rays, while its $gamma$-ray efficiency is consistent with an average efficiency for $gamma$-pulsars of similar age. Our optical flux upper limit for the pulsar is consistent with the long-wavelength extrapolation of its X-ray spectrum while the nebula flux upper limit does not constrain the respective extrapolation.
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