  Integrable or near-integrable magnetic fields are prominent in the design of
plasma confinement devices. Such a field is characterized by the existence of a
singular foliation consisting entirely of invariant submanifolds. A regular
leaf, known as a flux surface,of this foliation must be diffeomorphic to the
two-torus. In a neighborhood of a flux surface, it is known that the magnetic
field admits several exact, smooth normal forms in which the field lines are
straight. However, these normal forms break down near singular leaves including
elliptic and hyperbolic magnetic axes. In this paper, the existence of exact,
smooth normal forms for integrable magnetic fields near elliptic and hyperbolic
magnetic axes is established. In the elliptic case, smooth near-axis Hamada and
Boozer coordinates are defined and constructed. Ultimately, these results
establish previously conjectured smoothness properties for smooth solutions of
the magnetohydrodynamic equilibrium equations. The key arguments are a
consequence of a geometric reframing of integrability and magnetic fields; that
they are presymplectic systems.
