Killing tensors on projective spaces

A Killing tensor field on a Riemannian space corresponds to an integral of the geodesic flow polynomial in momenta. A (contravariant) Killing tensor field is called \emph{decomposable} if it is a polynomial in Killing vector fields. While all Killing tensor fields on the spaces of constant curvature and on the complex projective space are decomposable, there is an explicitly constructed family of indecomposable quadratic Killing tensor fields on the quaternionic projective spaces $\mathbb{H}P^n, \, n \ge 3$. We prove that the algebra of Killing tensor fields on the quaternionic projective space is generated by Killing vector fields and these indecomposable quadratic Killing tensor fields. We also give another proof of the fact that the algebra of Killing tensor fields on the complex projective space is generated by Killing vector fields.