Infinite-dimensional Hamiltonian systems appear naturally in the rich algebraic structure of symplectic field theory. Carefully defining a generalization of gravitational descendants and adding them to the picture, one can produce an infinite number of symmetries of such systems. As in Gromov-Witten theory, the study of the topological meaning of gravitational descendants yields new differential equations for the SFT Hamiltonian, where the key point is to understand the dependence of the algebraic constructions on choices of auxiliary data such as differential forms representing cohomology classes on the target and coherent collections of sections used to define gravitational descendants. © 2010 The Author(s). Published by Oxford University Press. All rights reserved.