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.
Integrable geodesic flow in 3D and webs of maximal rank
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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.