  We present a detailed characterization of planes of satellites in the Milky
Way (MW) and M31 systems. To this end we introduce an extension to the
'4-galaxy-normal density plot' method \citep{Pawlowski13}, by which plot
over-densities signal the normal direction to predominant planes of satellites
within a given sample. For a given over-density, the extension provides a
\textit{collection} of planes, each including a different number of objects
$N_{\rm sat}$. We apply this method to the position data of confirmed MW and
M31 satellites and quantify the quality of planes through the outputs of a
Tensor of Inertia plane-fitting technique. Plane quality is quantified in terms
of population ($N_{\rm sat}$) and flattening (the short-to-long axis ratio
$c/a$ or the rms thickness normal to the plane). Therefore, planes with the
same population or flattening can be compared with each other allowing us to
single-out best-quality planes. For the first time, we study the second-most
predominant planar configuration of satellites in M31, singling out a plane
with 18 satellite members that shows a quality comparable to the Great Plane of
Andromeda (GPoA, with $N_{\rm sat}=19$) despite it being more affected by
distance uncertainties. This structure is viewed nearly face-on from the MW and
is approximately normal to the GPoA. Overall, we find planes of satellites
around the MW and M31 with higher qualities than those previously reported with
a given $N_{\rm sat}$. We also show that mass plays no role in determining a
satellite's membership or not to the respective best-quality planes.
