Geodesic first integrals with explicit path-parameter dependence in Riemannian space–times

In a Riemannian space Vn general formulas are obtained for geodesic first integrals which are mth order polynomials in the tangent vector and which are assumed to depend explicitly on the path parameter s. It is found that such first integrals must also be polynomials in s. Necessary and sufficient conditions are found for the existence of these first integrals. The existence of many well-known symmetries such as homothetic motions (scale change), affine collineations, conformal motions, projective collineations, conformal collineations, or special curvature collineations are shown to be sufficient for the existence of such first integrals with explicit path-parameter dependence. To illustrate the theory, geodesic first integrals of this type have been calculated for four Riemannian space–times of general relativity.