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A system consists of two masses interacting with gravitational force, rotating around their centre of mass. If we only consider the $xy$ plane where the masses rotate, the system has 8 degrees of freedom ($xy$ position and velocity for each mass). How many independent constants of motion are there?

I know: (1) energy, (2, 3) momentum in $x$ and $y$, (4) angular momentum. So it probably has 4 or more constants of motion. Do I miss the others?

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