We consider complete integrability of the Hamiltonian of the geodesic flow of two particular solutions, the Kerr-Newman and the FLRW metrics of the Einstein equations in the sense of Liouville. We construct recursion operators using first integrals, and then obtain constants of motion of the geodesic flows by using the recursion operators.
Tsukasa Takeuchi. "On the Construction of Recursion Operators for the Kerr-Newman and FLRW Metrics." J. Geom. Symmetry Phys. 37 85 - 96, 2015. https://doi.org/10.7546/jgsp-37-2015-85-96