Canonical decomposition of projective and affine killing vectors on the tangent bundle

For an affine connection on the tangent bundle $T(M)$ obtained by lifting an affine connection on $M$, the structure of vector fields on $T(M)$ which generate local one-parameter groups of projective and affine collineations is described. On the $T(M)$ of a complete irreducible Riemann manifold, every projective collineation is affine. On the $T(M)$ of a projectively Euclidean space, every affine collineation preserves the fibration of $T(M)$, and on the $T(M)$ of a projectively non-Euclidean space which is maximally homogeneous (in the sense of affine collineations) there exist affine collineations permuting the fibers of $T(M)$.