  We present a detailed characterization of planes of satellite galaxies in the
Milky Way (MW) and M31. For a positional analysis, we introduce an extension to
the `4-galaxy-normal density plot' method \citep[][P13]{Pawlowski13}. It finds
the normal directions to the predominant planar configurations of satellites of
a system, yielding for each a \textit{collection} of planes of increasing
member satellites. This allows to quantify the quality of planes in terms of
population ($N_{\rm sat}$) and spatial flattening ($c/a$). We apply this method
to the latest data for confirmed MW and M31 satellite samples, with 46 and 34
satellites, respectively. New MW satellites form part of planes previously
identified from the sample with $N_{\rm sat}=27$ studied in P13: we identify a
new plane with $N_{\rm sat}=39$ as thin as the VPOS-3 ($c/a\sim 0.2$), and with
roughly the same normal direction; so far the most populated plane that thin
reported in the Local Group. We introduce a new method to determine, using
kinematic data, the axis of maximum co-orbitation of MW satellites.
Interestingly, this axis approximately coincides with the normal to the former
plane: $\geq45\pm5\%$ of satellites co-orbit. In M31 we discover a plane with
$N_{\rm sat}=18$ and $c/a\sim0.15$, i.e., quality comparable to the GPoA, and
perpendicular to it. This structure is viewed face-on from the Sun making it
susceptible to M31 satellite distance uncertainties. An estimation of the
perpendicular velocity dispersion suggests it is dynamically unstable. Finally
we find that mass is not a property determining a satellite's membership to
good quality planes.
