  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.
