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In this note, we study properties of the gradient map of the isoparametricpolynomial. For a given isoparametric hypersurface in sphere, we calculateexplicitly the gradient map of its isoparametric polynomial which turns outmany interesting phenomenons and applications. We find that it should map notonly the focal submanifolds to focal submanifolds, isoparametric hypersurfacesto isoparametric hypersurfaces, but also map isoparametric hypersurfaces tofocal submanifolds. In particular, it turns out to be a homogeneous polynomialautomorphism on certain isoparametric hypersurface. As an immediateconsequence, we get the Brouwer degree of the gradient map which was firstlyobtained by Peng and Tang with moving frame method. Following Farina'sconstruction, another immediate consequence is a counter example of theBr\'ezis question about the symmetry for the Ginzburg-Landau system indimension 6, which gives a partial answer toward the Open problem 2 raised byFarina.

Gradient map of isoparametric polynomial and its application to Ginzburg-Landau system