We analyzed a deep {it XMM-Newton} observation of the radio-quiet $gamma$-ray PSR J2055+2539. The spectrum of the X-ray counterpart is non-thermal, with a photon index of $Gamma$=2.36$pm$0.14 (1$sigma$ confidence). We detected X-ray pulsations with a pulsed fraction of (25$pm$3)% and a sinusoidal shape. Taking into account considerations on the $gamma$-ray efficiency of the pulsar and on its X-ray spectrum, we can infer a pulsar distance ranging from 450 pc to 750 pc. We found two different nebular features associated to PSR J2055+2539 and protruding from it. The angle between the two nebular main axes is $sim$ (162.8$pm$0.7)$^{circ}$. The main, brighter feature is 12 long and $<$20 thick, characterized by an asymmetry with respect to the main axis that evolves with the distance from the pulsar, possibly forming a helical pattern. The secondary feature is 250 $times$ 30. Both nebulae present an almost flat brightness profile with a sudden decrease at the end. The nebulae can be fitted either by a power-law model or a thermal bremsstrahlung model. A plausible interpretation of the brighter nebula is in terms of a collimated ballistic jet. The secondary nebula is most likely a classical synchrotron-emitting tail.
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