  In this article we study eternal solutions to the Allen-Cahn equation in the
3-sphere, in view of the connection between the gradient flow of the associated
energy functional, and the mean curvature flow. We construct eternal integral
Brakke flows that connect Clifford tori to equatorial spheres, and study a
family of such flows, in particular their symmetry properties. Our approach is
based on the realization of Brakke's motion by mean curvature as a singular
limit of Allen-Cahn gradient flows, as studied by Ilmanen, and Tonegawa, and it
uses the classification of ancient gradient flows in spheres, by K. Choi and C.
Mantoulidis, as well as the rigidity of stationary solutions with low Morse
index proved by F. Hiesmayr.
