  Every Killing tensor field on the space of constant curvature and on the
complex projective space can be decomposed into the sum of symmetric tensor
products of Killing vector fields (equivalently, every polynomial in the
velocities integral of the geodesic flow is a polynomial in the linear
integrals). This fact led to the natural question on whether this property is
shared by Killing tensor fields on all Riemannian symmetric spaces. We answer
this question in the negative by constructing explicit examples of quadratic
Killing tensor fields which are not quadratic forms in the Killing vector
fields on the quaternionic projective spaces $\mathbb{H} P^n, n \ge 3$, and on
the Cayley projective plane $\mathbb{O} P^2$.
