Robert, when I am saying "embedded" I am having in mind a bunch of very recent papers in Phys.Rev.D and in Comm.Math.Physics which demonstrated that such an embedding indeed can be accomplished. This is very interesting as far as I can see things. As for Schwarzschild space-mimes, reference  K.Brown, Reflections on Relativity,www.mathpages.com is claiming that Kepler's laws are compatible with this Schwarzshield space-time. Brown's calculations seem to me convincing. Yes standard Newton's mechanics can and is typically being used but if you take our Sun and 8 planets and try to embed this system into spaces of constant curvature, you would quickly recognize that this cannot be done. This is result by Florin Diacu. Thus, we have to make a choice once and for all between Einsteinian relativity and Newton's mechanics not from the point of computational convenience but from the point of general principle....

Not at all! Treating Solar system  withing GR-this is all what I am interested in.  If you are interested in references, please, contact me directly via [email protected] Thus far these are as puzzling for me as they are for you. Still, Bertrand spaces is interesting development in GR and, even in Newtonian mechanics.

Yes, Perlick's paper is the point of departure. I am at home now,while all papers are at work. I'll soon come to work and will be able to be more specific. In the meanwhile... From the top of my head I remember that subsequent papers were trying to connect the effects of dark matter and the motion of, say,planets, or galaxies, etc. I have not read yet a single paper, just collecting information.However, I do remember a paper in Phys.Rev.D (2013) in which they managed to connect the Bertrand and Schwarzschield space-times. It would be interesting for me to hear your comments on this paper. The whole purpose for doing all this is stemming from my paper  http://adsabs.harvard.edu/abs/2010arXiv1006.4650K   which I am now trying to update and to revise under pressure of circumstances. In this paper I was effectively using the Scwarzschield model  and now I've run into the Bertrand spaces and would like to understand what use I can make out of them in conjunction with what is in my paper.

Robert, now I am at work and can supply you the missing details. There are

1. Class.Quant.Gravity Vol.25 (2008) 165005 (Bertrand spacetimes as Kepler/oscillator potentials)

2.Comm.Math.Phys. 290 (2009) 1033-1049. 

3. Phys.Rev.D 88 (2013) 083532 (Astrophysics of bertrand space-times).

4. arXiv: 1408.6152 (Bertrand spaces and projected closed orbits in general relativity)

Surely, I am having much more.However, these are related directly to the content of our dialogue. It would be nice to hear your impressions. Warmest regards. Arkady

Robert,take your time. I've looked already at all of them and found only one use thus far. Bertrand spaces are seemingly of use (not counting their uses in  exactly integrable systems) in astrophysics, e.g. if one is interested in motions of  individual stars around galactic centers.  As authors claim, the Bertrrand-type metrics(I or II) then are better suited than the Schwarzshield -type metrics. 

I agree 100%. However, my immediate goals are more decent. Thus, at this moment I would rather settle for less. In any case, I was not expecting that I would reach these conclusions in about a day time or so. Therefore I've raised the issue in the first place. But , surely, it is always better to hear from somebody things which you know already. This is very reassuring.Thank you! I am also working on the problem (and reached some progress) about knotted configurations of gauge fields which serve as sources of gravity field. They also can carry charges, etc.