Does anyone know wether there exists an analytic solution for the
axisymmetric Eddington potential?

The best I could find in a short search on ADS

http://adsabs.harvard.edu/abstract_service.html

was a reference to this paper:


http://adsabs.harvard.edu/cgi-bin/np...1582 8ae11107

published in "Formation and evolution of galaxies and large structures
in the universe;"
Proceedings of the Third Moriond Astrophysics Meeting,
La Plagne, France, March 1983 (A84-45401 22-90).
Dordrecht, D. Reidel Publishing Co., 1984, p. 315-325.

The abstract states:

A technique for fitting an Eddington potential globally to the
potential from the equilibrium phase of a triaxial N-body model is
developed on the basis of the formalism of de Zeeuw (1984) and
Lynden-Bell (1962) and applied to the two-dimensional case. The results
are presented in graphs and found to indicate the applicability of the
model to astrophysically realistic models in three dimensions.

Good luck!