  In order to shed some light in the meaning of the relativistic multipolar
expansions we consider different static solutions of the axially symmetric
vacuum Einstein equations that in the non relativistic limit have same
Newtonian moments. The motion of test particles orbiting around different
deformed attraction centers with the same Newtonian limit is studied paying
special attention to the advance of the perihelion. We find discrepancies in
the fourth order of the dimensionless parameter (mass of the attraction
center)/(semilatus rectum). An evolution equation for the difference of the
radial coordinate due to the use of different general relativistic multipole
expansions is presented.
