2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94495In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time-dependent symmetries. In particular we describe how the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time-dependent Hamiltonian function. We also present a very general Lagrangian formulation of the procedure of reduction. As an application we consider the case of the Painleve' equations PI, PII, PIII, PVI and also certain higher order systems appeared in the theory of Frobenius manifolds.46 pages, Tex, to be published in "Differential Geometry and Applications"Exactly Solvable and Integrable Systems35Q53, 58F07, 70H33text