  It is known that corresponding to each isometry there exist a conserved
quantity. It is also known that the Lagrangian of the line element of a space
is conserved. Here we investigate the possibility of the existence of "new"
conserved quantities, i.e. other than the Lagrangian and associated with the
isometries, for spaces of different curvatures. It is found that there exist
new conserved quantities only for the spaces of zero curvature or having a
section of zero curvature.
