Exact and quasi-exact Newton dynamics of n − 1 and n bodies n − 1 of which have equal masses

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For n,n≥3 gravitating bodies n−1 of which have equal masses planar exact and spatial quasi-exact solutions of the Newton equation of motion are found. Exact planar solutions are also found for this equation for n−1,n≥3 bodies with equal masses. These solutions describe a motion of the bodies with equal masses in such a way that their coordinates coincide with vertices of a regular polygon in horizontal planes while the remaining body is either immobile at the origin or moves along the vertical axis crossing the origin. The Weinstein and center Lyapunov theorems are utilized.