  In this note, we study properties of the gradient map of the isoparametric
polynomial. For a given isoparametric hypersurface in sphere, we calculate
explicitly the gradient map of its isoparametric polynomial which turns out
many interesting phenomenons and applications. We find that it should map not
only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces
to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to
focal submanifolds. In particular, it turns out to be a homogeneous polynomial
automorphism on certain isoparametric hypersurface. As an immediate
consequence, we get the Brouwer degree of the gradient map which was firstly
obtained by Peng and Tang with moving frame method. Following Farina's
construction, another immediate consequence is a counter example of the
Br\'ezis question about the symmetry for the Ginzburg-Landau system in
dimension 6, which gives a partial answer toward the Open problem 2 raised by
Farina.
