  We search for vast planes of satellites (VPoS) in a high resolution
simulation of the Local Group performed by the CLUES project, which improves
significantly the resolution of former similar studies. We use a simple method
for detecting planar configurations of satellites, and validate it on the known
plane of M31. We implement a range of prescriptions for modelling the satellite
populations, roughly reproducing the variety of recipes used in the literature,
and investigate the occurence and properties of planar structures in these
populations. The structure of the simulated satellite systems is strongly
non-random and contains planes of satellites, predominantly co-rotating, with,
in some cases, sizes comparable to the plane observed in M31 by Ibata et al..
However the latter is slightly richer in satellites, slightly thinner and has
stronger co-rotation, which makes it stand out as overall more exceptional than
the simulated planes, when compared to a random population. Although the
simulated planes we find are generally dominated by one real structure, forming
its backbone, they are also partly fortuitous and are thus not kinematically
coherent structures as a whole. Provided that the simulated and observed planes
of satellites are indeed of the same nature, our results suggest that the VPoS
of M31 is not a coherent disc and that one third to one half of its satellites
must have large proper motions perpendicular to the plane.
