  We use the theory of isoparametric functions to investigate gradient Ricci
solitons with constant scalar curvature. We show rigidity of gradient Ricci
solitons with constant scalar curvature under some conditions on the Ricci
tensor, which are all satisfied if the manifold is curvature homogeneous. This
leads to a complete description of four- and six-dimensional Kaehler gradient
Ricci solitons with constant scalar curvature.
