  In recent works we have constructed axisymmetric solutions to the
Euler-Poisson equations which give mathematical models of slowly uniformly
rotating gaseous stars. We try to extend this result to the study of solutions
of the Einstein-Euler equations in the framework of the general theory of
relativity. Although many interesting studies have been done about axisymmetric
metric in the general theory of relativity, they are restricted to the region
of the vacuum. Mathematically rigorous existence theorem of the axisymmetric
interior solutions of the stationary metric corresponding to the
energy-momentum tensor of the perfect fluid with non-zero pressure may be not
yet established until now except only one found in the pioneering work by U.
Heilig done in 1993. In this article, along a different approach to that of
Heilig's work, axisymmetric stationary solutions of the Einstein-Euler
equations are constructed near those of the Euler-Poisson equations when the
speed of light is sufficiently large in the considered system of units, or,
when the gravitational field is sufficiently weak.
