  We study the Hamiltonian formulation for a parametrized electromagnetic field
with the purpose of clarifying the interplay between parametrization and gauge
symmetries. We use a geometric approach which is tailor-made for theories where
embeddings are part of the dynamical variables. Our point of view is global and
coordinate free. The most important result of the paper is the identification
of sectors in the primary constraint submanifold in the phase space of the
model where the number of independent components of the Hamiltonian vector
fields that define the dynamics changes. This explains the non-trivial behavior
of the system and some of its pathologies.
