Embedding and splitting ordinary differential equations in normal form

We introduce an embedding of real or complex n-dimensional space Kn as an algebraic variety V which is determined by the action of a linear one-parameter group. Every analytic vector field on Kn corresponds to some embedded vector field on V. For a symmetric vector field this embedded vector field splits into a reduced system and a direct sum of non-autonomous linear systems. Examples and applications are mostly concerned with Poincaré–Dulac normal forms. Embeddings provide a natural setting for perturbations of symmetric systems, in particular of systems in normal form up to some degree.