  We consider the equations for the coefficients of stationary rotating
axisymmetric metrics governed by the Einstein-Euler equations, that is, the
Einstein equations together with the energy-momentum tensor of a barotropic
perfect fluid. Although the reduced system of equations for the potentials in
the co-rotating co-ordinate system is known, we derive the system of equations
for potentials in the so called zero angular momentum observer co-ordinate
system. We newly give a proof of the equivalence between the reduced system and
the full system of Einstein equations. It is done under the assumption that the
angular velocity is constant on the support of the density. Also the
consistency of the equations of the system is analyzed. On this basic theory we
construct on the whole space the stationary asymptotically flat metric
generated by a slowly rotating compactly supported perfect fluid with vacuum
boundary.
