Reduction and integrability: a geometric perspective

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.