Killing tensors on reducible spaces

We prove that on the product of two Riemannian manifolds one of which is compact, any Killing tensor is reducible, that is, is the sum of products of Killing tensors on the factors. The same is true for the lifts to the universal cover of Killing tensors on a compact manifold with reducible holonomy. We give a local description of Killing tensors on product manifolds and present an example of a complete product manifold whose factors are locally irreducible which admits an irreducible Killing tensor field.