  It is well-known that in any codimension a simply connected Euclidean minimal
surface has an associated one-parameter family of minimal isometric
deformations. In this paper, we show that this is just a special case of the
associated family to any simply connected elliptic surface for which all
curvature ellipses of a certain order are circles. We also provide the
conditions under which this associated family is trivial, extending the known
result for minimal surfaces. As an application, we show how the associated
family of a minimal Euclidean submanifold of rank two is determined by the
associated family of an elliptic surface clarifying the geometry around the
associated family of these higher dimensional submanifolds.
