Invariant theory without groups

In talk I will discuss the following result obtained jointly with M. Radeschi. Any equidistant decompostion of a  unit sphere is defined by polynomial equations and gives rise to  a finitely generated algebra
of polynomials. In the case the equidistance decomposition is given by an action of an orthogonal group, this is just the algebra of invariants of the corresponding representation. However, there are huge families of non-homogeneous equidistant decomspositions, for instance given by isoparametric hypersurfaces or
representations of Clifford algebras. On the one hand this defines new interesting finitely generated algebras of polynomials, on the other hand there is some hope to use algebraic tools to investigate the  geometry and topology of the foliation.