  We show that the conservation laws for the geodesic equation which are
associated to affine symmetries can be obtained from symmetries of the
Lagrangian for affinely parametrized geodesics according to Noether's theorem,
in contrast to claims found in the literature. In particular, using Aminova's
classification of affine motions of Lorentzian manifolds, we show in detail how
affine motions define generalized symmetries of the geodesic Lagrangian. We
compute all infinitesimal proper affine symmetries and the corresponding
geodesic conservation laws for all homogeneous solutions to the Einstein field
equations in four spacetime dimensions with each of the following
energy-momentum contents: vacuum, cosmological constant, perfect fluid, pure
radiation, and homogeneous electromagnetic fields.
