  Time-domain analysis of an archival XMM-Newton observation unveiled a very
unusual variability pattern in the soft X-ray emission of PSR J1311-3430, a
black widow millisecond pulsar in a tight binary (P_B=93.8 min) with a very
low-mass (M~0.01 Msun) He companion star, known to show flaring emission in the
optical and in the X-rays. A series of six pulses with a regular recurrence
time of ~124 min is apparent in the 0.2-10 keV light curve of the system, also
featuring an initial, bright flare and a quiescent phase lasting several hours.
The X-ray spectrum does not change when the pulses are seen and is consistent
with a power law with photon index Gamma~1.6, also describing the quiescent
emission. The peak luminosity of the pulses is of several 10^32 erg/s.
Simultaneous observations in the U band with the Optical Monitor onboard XMM
and in the g' band from the Las Cumbres Observatory do not show any apparent
counterpart of the pulses and only display the well-known orbital modulation of
the system. We consider different hypotheses to explain the recurrent pulses:
we investigate their possible analogy with other phenomena already observed in
this pulsar and in similar systems and we also study possible explanations
related to the interaction of the energetic pulsar wind with intrabinary
material, but we found none of these pictures to be convincing. We identify
simultaneous X-ray observations and optical spectroscopy as a possible way to
constrain the nature of the phenomenon.
