Integrability of geodesic motions in curved manifolds through non-local conserved charges

In this work we study the general system of geodesic equations for the case of a massive particle moving on an arbitrary curved manifold. The investigation is carried out from the symmetry perspective. By exploiting the parametrization invariance property of the system we define nonlocal conserved charges that are independent from the typical integrals of motion constructed out of possible Killing vectors/tensors of the background metric. We show that with their help every two dimensional surface can - at least in principle - be characterized as integrable. Due to the nonlocal nature of these quantities not more than two can be used at the same time unless the solution of the system is known. We demonstrate that even so, the two dimensional geodesic problem can always be reduced to a single first order ordinary differential equation; we also provide several examples of this process.