A surprising fibration of S3 x S3 by great 3-spheres

In this paper, we describe a new surprising example of a fibration of the Clifford torus S3 x S3 in the 7-sphere by great 3-spheres, which is fiberwise homogeneous but whose fibers are not parallel to one another. In particular it is not part of a Hopf fibration. A fibration is fiberwise homogeneous when for any two fibers there is an isometry of the total space taking fibers to fibers and taking the first given fiber to the second one. We also describe in detail the geometry of this surprising fibration and how it differs from the Hopf fibration.