  In this paper we investigate $m$-dimensional complete minimal submanifolds in
Euclidean spheres with index of relative nullity at least $m-2$ at any point.
These are austere submanifolds in the sense of Harvey and Lawson \cite{harvey}
and were initially studied by Bryant \cite{br}. For any dimension and
codimension there is an abundance of non-complete examples fully described by
Dajczer and Florit \cite{DF2} in terms of a class of surfaces, called elliptic,
for which the ellipse of curvature of a certain order is a circle at any point.
Under the assumption of completeness, it turns out that any submanifold is
either totally geodesic or has dimension three. In the latter case there are
plenty of examples, even compact ones. Under the mild assumption that the
Omori-Yau maximum principle holds on the manifold, a trivial condition in the
compact case, we provide a complete local parametric description of the
submanifolds in terms of $1$-isotropic surfaces in Euclidean space. These are
the minimal surfaces for which the standard ellipse of curvature is a circle at
any point. For these surfaces, there exists a Weierstrass type representation
that generates all simply-connected ones.
