  We use an isomorphism between the space of valence two Killing tensors on an
n-dimensional constant sectional curvature manifold and the irreducible
GL(n+1)-representation space of algebraic curvature tensors in order to
translate the Nijenhuis integrability conditions for a Killing tensor into
purely algebraic integrability conditions for the corresponding algebraic
curvature tensor, resulting in two simple algebraic equations of degree two and
three. As a first application of this we construct a new family of integrable
Killing tensors.
