  In this paper, we examine a static gravitational field with axial symmetry
over probe particles in the solar system. Using the Weyl conformastatic
solution as a model, we find a non-standard expression to perihelion advance
due to the constraints imposed by the topology of the local gravitational
field. We show that the application of the slow motion condition to the
geodesic equations without altering Einstein's equations does not necessarily
lead to the Newtonian limit; rather it leads to an intermediate
nearly-newtonian gravitational stage, which can be applied to astrophysical
problems in solar scale. We apply the model to the perihelion advance of inner
planets and minor objects (NEO asteroids and the comets). As a result, we
obtain an expression of a non-standard relativistic precession that reveals a
close agreement to observational data calibrated with the Ephemerides of the
Planets and the Moon (EPM2011). The study of perihelion advance of eight small
celestial bodies (asteroids and comets) is also considered.
