n functionally independent commutative quadratic integrals for a geodesic flow that are simultaneously diagonalisable imply the metric comes from the Stäckel construction and admits orthogonal separation of variables.
M´ emoire sur l’int´ egration des ´ equations diff´ erentielles du mouvement d’un nombre quelconque de points mat´ eriels
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Integrable geodesic flows with simultaneously diagonalisable quadratic integrals
n functionally independent commutative quadratic integrals for a geodesic flow that are simultaneously diagonalisable imply the metric comes from the Stäckel construction and admits orthogonal separation of variables.