  The balanced configurations are those n-body configurations which admit a
relative equilibrium motion in a Euclidean space E of high enough dimension 2p.
They are characterized by the commutation of two symmetric endomorphisms of the
(n-1)-dimensional Euclidean space of codispositions, the intrinsic inertia
endomorphism B which encodes the shape and the Wintner-Conley endomorphism A
which encodes the forces. In general, p is the dimension d of the
configuration, which is also the rank of B. Lowering to 2(d-1) the dimension of
E occurs when the restriction of A to the (invariant) image of B possesses a
double eigenvalue. It is shown that, while in the space of all dxd-symmetric
endomorphisms, having a double eigenvalue is a condition of codimension 2 (the
avoided crossings of physicists), here it becomes of codimension 1 provided
some condition (H) is satisfied. As the condition is always satisfied for
configurations of the maximal dimension (i.e. if d=n-1), this implies in
particular the existence, in the neighborhood of the regular tetrahedron
configuration of 4 bodies with no three of the masses equal, of exactly 3
families of balanced configurations which admit relative equilibrium motion in
a four dimensional space.
