  This paper is concerned exclusively with axisymmetric spacetimes. We want to
develop reductions of Einstein's equations which are suitable for numerical
evolutions. We first make a Kaluza-Klein type dimensional reduction followed by
an ADM reduction on the Lorentzian 3-space, the (2+1)+1 formalism. We include
also the Z4 extension of Einstein's equations adapted to this formalism. Our
gauge choice is based on a generalized harmonic gauge condition. We consider
vacuum and perfect fluid sources.
  We use these ingredients to construct a strongly hyperbolic first-order
evolution system and exhibit its characteristic structure. This enables us to
construct constraint-preserving stable outer boundary conditions. We use
cylindrical polar coordinates and so we provide a careful discussion of the
coordinate singularity on axis. By choosing our dependent variables
appropriately we are able to produce an evolution system in which each and
every term is manifestly regular on axis.
