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We make an observation about Galilean transformation on a 1-D mass variable system which leads us to the right way to deal with mass variable systems. Then using this observation, we study two-body gravitational problem where the mass of one of the bodies varies and suffers a damping-antidamping effect due to star wind during its motion. For this system, a constant of motion, a Lagrangian and a Hamiltonian are given for the radial motion, and the period of the body is studied using the constant of motion of the system. Our theoretical results are applied to Halley’s Comet.