  Axially symmetric spacetimes are the only models for isolated systems with
continuous symmetries that also include dynamics. For such systems, we review
the reduction of the vacuum Einstein field equations to their most concise form
by dimensionally reducing to the three-dimensional space of orbits of the
Killing vector, followed by a conformal rescaling. The resulting field
equations can be written as a problem in three-dimensional gravity with a
complex scalar field as source. This scalar field, the Ernst potential is
constructed from the norm and twist of the spacelike Killing field. In the case
where the axial Killing vector is twist-free, we discuss the properties of the
axis and simplify the field equations using a triad formalism. We study two
physically motivated triad choices that further reduce the complexity of the
equations and exhibit their hierarchical structure. The first choice is adapted
to a harmonic coordinate that asymptotes to a cylindrical radius and leads to a
simplification of the three-dimensional Ricci tensor and the boundary
conditions on the axis. We illustrate its properties by explicitly solving the
field equations in the case of static axisymmetric spacetimes. The other choice
of triad is based on geodesic null coordinates adapted to null infinity as in
the Bondi formalism. We then explore the solution space of the twist-free
axisymmetric vacuum field equations, identifying the known (unphysical)
solutions together with the assumptions made in each case. This singles out the
necessary conditions for obtaining physical solutions to the equations.
