  A theorem is derived which determines higher order first integrals of
autonomous holonomic dynamical systems in a general space, provided the
collineations and the Killing tensors -- up to the order of the first integral
-- of the kinetic metric, defined by the kinetic energy of the system, can be
computed. The theorem is applied in the case of Newtonian autonomous
conservative dynamical systems of two degrees of freedom, where known and new
integrable and superintegrable potentials that admit cubic first integrals are
determined.
