  The confinement of most satellite galaxies in the Local Group to thin planes
presents a challenge to the theory of hierarchical galaxy clustering. The
PAndAS collaboration has identified a particularly thin configuration with
kinematic coherence among companions of M31 and there have been long standing
claims that the dwarf companions to the Milky Way lie in a plane roughly
orthogonal to the disk of our galaxy. This discussion investigates the possible
origins of four Local Group planes: the plane similar, but not identical to
that identified by PAndAS, an adjacent slightly tilted plane, and two planes
near the Milky Way: one with nearer galaxies and the other with more distant
ones. Plausible orbits are found by using a combination of Numerical Action
methods and a backward in time integration procedure. For M31, M33, IC10, and
LeoI, solutions are found that are consistent with measurements of their proper
motions. For galaxies in planes, there must be commonalities in their proper
motions, and this constraint greatly limits the number of physically plausible
solutions. Key to the formation of the planar structures has been the
evacuation of the Local Void and consequent build-up of the Local Sheet, a wall
of this void. Most of the M31 companion galaxies were born in early-forming
filamentary or sheet-like substrata that chased M31 out of the void. M31 is a
moving target because of its attraction toward the Milky Way, and the result
has been alignments stretched toward our galaxy. In the case of the
configuration around the Milky Way, it appears that our galaxy was in a
three-way competition for companions with M31 and Centaurus A. Only those
within a modest band fell our way. The Milky Ways' attraction toward the Virgo
Cluster resulted in alignments along the Milky Way-Virgo Cluster line.
