  Reduction theory has played a major role in the study of Hamiltonian systems.
On the other hand, the Hamilton-Jacobi theory is one of the main tools to
integrate the dynamics of certain Hamiltonian problems and a topic of research
on its own. Moreover, the construction of several symplectic integrators rely
on approximations of a complete solution of the Hamilton-Jacobi equation. The
natural question that we address in this paper is how these two topics
(reduction and Hamilton-Jacobi theory) fit together. We obtain a reduction and
reconstruction procedure for the Hamilton-Jacobi equation with symmetries, even
in a generalized sense to be clarified below. Several applications and
relations to other reductions of the Hamilton-Jacobi theory are shown in the
last section of the paper. It is remarkable that as a by-product we obtain a
generalization of the Ge-Marsden reduction procedure. Quite surprinsingly, the
classical ansatzs available in the literature to solve the Hamilton-Jacobi
equation are also particular instances of our framework.
