  A geometric approach to integrability and reduction of dynamical system is
developed from a modern perspective. The main ingredients in such analysis are
the infinitesimal symmetries and the tensor fields that are invariant under the
given dynamics. Particular emphasis is given to the existence of invariant
volume forms and the associated Jacobi multiplier theory, and then the Hojman
symmetry theory is developed as a complement to Noether theorem and non-Noether
constants of motion. The geometric approach to Hamilton-Jacobi equation is
shown to be a particular example of the search for related field in a lower
dimensional manifold.
