Here's a proof that 2 planets orbiting each other will move on a planepic.twitter.com/kYqewE38Gw
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Actually this comes directly from the symmetry of the problem, which is a very fundamental property. It then works not only with Newtonian gravity but also in General Relativity. The conservation laws (as angular momentum) come from the underlying symmetries.
Both are right, angular momentum conservation is precisely a statement about symmetries (Noether’s theorem) We just view symmetries as ‘more fundamental’
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Actually for multiple objects it's really difficult (/ impossible) to express an exact solution in general relativity, hence to find the conserved quantities. For example energy is not conserved as the objects might generate gravitational waves.
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For a planar symmetry like this, the thing which is conserved is the orientation of the angular momentum vector, but not necessarily its value.
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