  We consider the 3-dimensional formulation of Einstein's theory for spacetimes
possessing a non-null Killing field $\xi^a$. It is known that for the vacuum
case some of the basic field equations are deducible from the others. It will
be shown here how this result can be generalized for the case of essentially
arbitrary matter fields. The systematic study of the structure of the
fundamental field equations is carried out. In particular, the existence of
geometrically preferred reference systems is shown. Using local coordinates of
this type two approaches are presented resulting resolvent systems of partial
differential equations for the basic field variables. Finally, the above
results are applied for perfect fluid spacetimes describing possible
equilibrium configurations of relativistic dissipative fluids.
