  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.
