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In the present paper, we study the three-body problem in the case where two bodies have equal masses, which imply the existence of a manifold of symmetric motions. We find conditions of existence of bounded symmetric motions. These conditions can be useful for elucidating those key circumstances that cause the existence of oscillating final evolutions. For the analysis of boundedness of motions, both the structure of the manifold of symmetrical motions and the integrals of energy and angular momentum are essential.

On Boundedness of Symmetrical Motions in the Three-Body Problem