THE TALE OF THE TWO TAILS OF THE OLDISH PSR J2055+2539

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.