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We use an isomorphism between the space of valence two Killing tensors on ann-dimensional constant sectional curvature manifold and the irreducibleGL(n+1)-representation space of algebraic curvature tensors in order totranslate the Nijenhuis integrability conditions for a Killing tensor intopurely algebraic integrability conditions for the corresponding algebraiccurvature tensor, resulting in two simple algebraic equations of degree two andthree. As a first application of this we construct a new family of integrableKilling tensors.

Algebraic Integrability Conditions for Killing Tensors on Constant Sectional Curvature Manifolds