  We study isoparametric submanifolds of rank at least two in a separable
Hilbert space, which are known to be homogeneous by a result of Heintze and
Liu, and associate to such a submanifold M and a point x in M a canonical
homogeneous structure (a certain bilinear map on the tangent space). We prove
that the homogeneous structure together with the second fundamental form
encodes all the information about M, and deduce from this the rigidity result
that M is completely determined by the second fundamental form and its
covariant derivative, thereby making such submanifolds accessible to
classification. As an essential step, we show that the one-parameter groups of
isometries constructed by Heintze and Liu to prove their homogeneity induce
smooth and hence everywhere defined Killing fields, implying the continuity of
the homogeneous structure. Here an important tool is the introduction of affine
root systems of isoparametric submanifolds.
