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Reduction theory has played a major role in the study of Hamiltonian systems. Whilst 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 relies 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 reduction of the Hamilton–Jacobi theory are shown in the last section of the paper. It is remarkable that as by-product we obtain a generalization of the Ge–Marsden reduction procedure [18] and the results in [17] . Quite surprisingly, the classical ansätze available in the literature to solve the Hamilton–Jacobi equation (see [2,19] ) are also particular instances of our framework. [ABSTRACT FROM AUTHOR]

We develop a Hamilton-Jacobi theory for singular Lagrangian systems in the Skinner-Rusk formalism. Comparisons with the Hamilton-Jacobi problem in the Lagrangian and Hamiltonian settings are discussed. [ABSTRACT FROM AUTHOR]