The maximum of potential energy and symmetries for massive galaxies and stars

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The consideration of the N-body gravitational problem equations can give to us some class of boundary-value problems defined on the "beem's" construction. One can considere it as weak or so-called finite element method's approximation with the linear function's basis. So there are time and coordinate dependent beems as distances between points and unknown coordinate functions which are defined on the beems. The possible sample from this class is simple Lalplas equation where potential energy are maximal. Some computer simulations can give to us symmetries on spherical coordinates (similar for well known mathematical structures configurations like Pascal or Desargus configurations).But in other hand, the consideration of the natural mass distribution can give to us evidence of the physical reality of the similar structures on the star sphere. Moreover, there are coupled invariancy for geometry and arithmetics.