We study the structure of those vector fields on the tangent bundle of an arbitrary smooth manifold which commute with the geodesic vector field defined by an affine connection. The study is restricted to polylinear fields generated by a pair of symmetric pseudotensor fields of type (k, 1) and (k + 1, 1), k ≥ 0, defined on the manifold. We establish an isomorphism between the space of infinitesimal automorphisms of fixed type and the space Heng hooktop sign _k of the solutions of a partial differential equation generalizing the Jacobi equation for the infinitesimal automorphisms of the connection. It is shown that the spaces Heng hooktop sign_k are finite-dimensional and form a graduated Lie algebra η = ⊕_k=0^∞ Heng hooktop sign_k. These algebras are classified in the case of one-dimensional manifolds. It is proved that if the geodesic vector field is complete, then so are the automorphisms corresponding to covariant constant fields of type (1, 1).