  We investigate whether there may be one or many companions orbiting at close
distance to the light cylinder around the extremely intermittent pulsars PSR
B1931+24 and PSR J1841-0500. These pulsars, behaving in a standard way when
they are active, also "switch off" for durations of several days, during which
their magnetospheric activity is interrupted or reduced. We constrained our
analysis on eight fundamental properties of PSR B1931+24 that summarise the
observations. We considered that the disruption/activation of the
magnetospheric activity would be caused by the direct interaction of the star
with the Alfv\'en wings emanating from the companions. We also considered the
recurrence period of 70 days to be the period of precession of the periastron
of the companions orbit. We analysed in which way the time scale of the
"on/off" pseudo-cycle would be conditioned by the precession of the periastron
and not by the orbital time scale, and we derived a set of orbital constraints
that we solved. We then compared the model, based on PSR 1931+24, with the
known properties of PSR 1841+0500. We conclude that PSR B1931+24 may be
surrounded at a close distance to the star by a stream of small bodies of
kilometric or sub-kilometric sizes that could originate from the tidal
disruption of a body of moderate size that fell at a close distance to the
neutron star on an initially very eccentric orbit. This scenario is also
compatible with the properties of PSR J1841-0500, although the properties of
PSR J1841-0500 are, by now, less constrained. These results raise new
questions. Why are the asteroids not yet evaporated ? What kind of interaction
can explain the disruption of the magnetospheric activity ? These questions are
the object of two papers in preparation that will complete the present
analysis.
