It seems to me, Saari's "spiderweb" exact solutions may be mathematically interesting. This is in spite of any complaints I have re Saari's hype about dark matter, and his enormous misuse of the word "stability." Why are they interesting? First of all, if you have some matter, and it is subject only to Newton's laws, and it has total momentum=0 and nonzero total angular momentum=A, then if you make the matter lose energy subject to those 2 constraints, it will eventually reach a state where all the matter lies in one plane (orthogonal to the A vector). So "discs" shaped much like many galaxies, and "saturn ring" configurations, thus arise naturally. I think the importance of discs was first realized by P.S.Laplace (?). The Saari spiderwebs are exact solutions, and they do lie in a plane. So in that sense they seem realistic. Second, the Saari spiderweb rotates as a rigid body. This is completely unlike actual galaxies and saturnian rings. Actual disc galaxies rotate with fairly "flat velocity curves," meaning the velocity v of the rotating stuff (which lies in the plane and orthogonal to the radial vector) stays approximately constant as its radial distance to the center changes. This empirical fact, first highly publicized by astronomers Vera Rubin and W.Kent Ford in the 1970s, is attributed to a conjectural halo of invisible "dark matter." A rigid body, on the other hand, would have linear velocity curve v=const*r. And a uniform matter density in the plane, would have a velocity curve v(r) obeying const*r^2 / r^2 = v^2 / r i.e. a "square root curve" v = const*r^(1/2). [If the matter in the plane were to have an areal density(r) correct to cause a flat velocity curve then that density presumably would need to obey const/r^2 * integral(0 to r) density(r)*r*dr = v^2 / r i.e. the density would be density = const/r.] And finally, the Saturn rings should have velocity curve obeying v^2 / r = const / r^2 i.e. v = const*r^(-1/2) the inverse-square-root curve. So the collection of powers of r we have here is {-1/2, 0 1/2, 1} for velocity curves v=const*r^P. Henry Baker has asked: "what would Saari's exact solutions do in general relativity rather than Newton?" Or about GR and galaxies. Normally GR is considered pretty irrelevant to galaxies because almost all motions are way slower than lightspeed, and the Einstein cosmical constant is too small to affect things on galactic length scales, so Newton works quite well. I would presume that exact solutions of GR much like Saari spiderweb, should exist, but nobody ever found them and they are probably not expressible in close form. However, there are a sequence of better and better approximations to GR, called the "post-Newtonian" and "post-Minkowskian" approximations. The first is Newton's laws. I believe the second is used by NASA to simulate the solar system. It seems to me that within any one of those approximations, exact Saari spiderweb solutions should exist, and they should exhibit essentially the same degree of difficulty to find them numerically, as it is to find Saari's Newton-law spiderweb exact solutions numerically. Another interesting remark is this. Among all spiderweb solutions, some small subset of them ought to be distinguished as "optimum" in some respect. What kinds of optimality would you want, and what spiderwebs would result? Finally, there are some unusual galaxies out there. Take a look at "Hoag's object" https://en.wikipedia.org/wiki/Hoag%27s_Object ---- PS. The "bullet cluster" although it has been hyped as a "proof" of the existence of dark matter, can also be (and has been) argued as proving the quantitative failure of the dark matter model. So be careful, there is a lot of hype about dark matter supported by very shallow and poor analyses.