  We establish the existence of an optimal partition for the Yamabe equation in
the whole space made up of mutually linearly isometric sets, each of them
invariant under the action of a group of linear isometries. To do this, we
establish the existence of a solution to a weakly coupled competitive Yamabe
system, whose components are invariant under the action of the group, and each
of them is obtained from the previous one by composing it with a linear
isometry. We show that, as the coupling parameter goes to minus infinity, the
components of the solutions segregate and give rise to an optimal partition
that has the properties mentioned above. Finally, taking advantage of the
symmetries considered, we establish the existence of infinitely many
sign-changing solutions for the Yamabe equation that are different from those
previously found in the by W.Y. Ding, and del Pino, Musso, Pacard and Pistoia
