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We analyzed two Chandra observations of PSR J2055+2539 (for a totalintegration time of $\sim$130 ks) in order to measure its proper motion andstudy its two elongated nebular features. We did not detect the proper motion,setting an upper limit of 240 mas yr$^{-1}$ (3$\sigma$ level), that translatesinto an upper limit on the transverse velocity of $\sim$700 km s$^{-1}$, for anassumed distance of 600 pc. A deep H$\alpha$ observation did not reveal thebow-shock associated with a classical pulsar wind nebula, thus precluding anindirect measurement of the proper motion direction. We determined the mainaxes of the two nebulae, which are separated by an angle of 160.8$^{\circ} \pm0.7^{\circ}$, using a new approach based on the Rolling Hough Transformation(RHT). We analyzed the shape of the first 8' (out of the 12' seen byXMM-Newton) of the brighter, extremely collimated one. Based on a combinationof our results from a standard analysis and a nebular modeling obtained fromthe RHT, we find that the brightest nebula is curved on an arcmin-scale, with athickness ranging from $\sim9$" to $\sim31$" and a possible (single ormultiple) helicoidal pattern. We could not constrain the shape of the fainternebula. We discuss our results in the context of other known similar features,with particular emphasis on the Lighthouse nebula (associated with PSRJ1101$-$6101). We speculate that a peculiar geometry of the powering pulsar mayplay an important role in the formation of such features.

The two tails of PSR J2055+2539 as seen by Chandra: analysis of the nebular morphology and pulsar proper motion