  In this paper we construct an end of a self-similar shrinking solution of the
mean curvature flow asymptotic to an isoparametric cone C and lying outside of
C. We call a cone C in $R^{n+1}$ an isoparametric cone if C is the cone over a
compact embedded isoparametric hypersurface $\Gamma \subset S^n$. The theory of
isoparametic hypersurfaces is extremely rich and there are infinitely many
distinct classes of examples, each with infinitely many members.
