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 \(