  For the gravitational $n$-body problem, the simplest motions are provided by
those rigid motions in which each body moves along a Keplerian orbit and the
shape of the system is a constant (up to rotations and scalings) configuration
featuring suitable properties. While in dimension $d \leq 3$ the configuration
must be central, in dimension $d \geq 4$ new possibilities arise due to the
complexity of the orthogonal group, and indeed there is a wider class of
$S$-balanced configurations, containing central ones, which yield simple
solutions of the $n$-body problem. Starting from recent results of the first
and third authors, we study the existence of continua of bifurcations branching
from a trivial branch of collinear $S$-balanced configurations and provide an
estimate from below on the number of bifurcation instants. In the last part of
the paper, by using the continuation method, we explicitly display the
bifurcation branches in the case of the three body problem for different
choices of the masses.
