  We study symmetric Killing 2-tensors on Riemannian manifolds and show that
several additional conditions can be realised only for Sasakian manifolds and
Euclidean spheres. In particular we show that (three)-Sasakian manifolds can
also be characterized by properties of the symmetric products of their
characteristic 1-forms. Moreover, we recover a result of S.~Gallot on the
characterization of spheres by means of functions satisfying a certain
differential equation of order three.
