Isoparametric hypersurfaces in Damek–Ricci spaces

We construct uncountably many isoparametric families of hypersurfaces in Damek–Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kähler angle. It follows that, in general, these examples are inhomogeneous and have nonconstant principal curvatures.We also find new cohomogeneity one actions on quaternionic hyperbolic spaces, and an isoparametric family of inhomogeneous hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane.