  The Lie symmetries of the geodesic equations in a Riemannian space are
computed in terms of the special projective group and its degenerates (affine
vectors, homothetic vector and Killing vectors) of the metric. The Noether
symmetries of the same equations are given in terms of the homothetic and the
Killing vectors of the metric. It is shown that the geodesic equations in a
Riemannian space admit three linear first integrals and two quadratic first
integrals. We apply the results in the case of Einstein spaces, the
Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each
case the Lie and the Noether symmetries are computed explicitly together with
the corresponding linear and quadratic first integrals.
