  We consider periodic and quasi-periodic solutions of the three-body problem
with homogeneous potential from the point of view of the equivariant calculus
of variations. First, we show that symmetry groups of the Lagrangian action
functional can be reduced to groups in a finite explicitly given list, after a
suitable change of coordinates. Then, we show that local symmetric minimizers
are always collisionless, without any assumption on the group other than the
fact that collisions are not forced by the group itself. Moreover, we describe
some properties of the resulting symmetric collisionless minimizers (Lagrange,
Euler, Hill-type orbits and Chenciner--Montgomery figure-eight).
