  The periastron position advance for geodesic motion in axially symmetric
solutions of the Einstein field equations belonging to the Weyl class of vacuum
solutions is investigated. Explicit examples corresponding to either static
solutions (single Chazy-Curzon, Schwarzschild and a pair of them), or
stationary solution (single rotating Chazy-Curzon and Kerr black hole) are
discussed. The results are then applied to the case of S2-SgrA$^*$ binary
system of which the periastron position advance will be soon measured with a
great accuracy.
