A system of differential equations in normal form near a rest point is often observed to decouple, or partially decouple, revealing various preserved geometrical structures such as fibrations and foliations. (A foliation is {\it preserved} if two orbits beginning on the same leaf share the same leaf at all times as they evolve; the leaves themselves need not be invariant.) It is often possible to see, in advance of computing the normal form, what these preserved structures are going to be. These structures are often not sufficient to render the normalized system analytically solvable, which raises the question: can these structures nevertheless be exploited to advantage when carrying out numerical solution of the equations?