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I am not a physicist.

Suppose there are two massive bodies orbiting each other. Their orbit will presumably define a plane. As far as I know, the existing solutions to the 3-body problem are limited to a plane.

Has any work been done on 3-dimensional orbits? For example two bodies are orbiting one another and a third body enters the system at right angle to their plane?

Specifically (1) Are there any solutions even for special cases of this? (2) Would such 3-D systems inevitability decay into 2 dimensions?

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Are all solutions of the 3-body problem restricted to 2 dimensions?

The five stationary solutions to the three body problem are restricted to a plane. That does not mean that all solutions are restricted to a plane.

Has any work been done on 3-dimensional orbits?

NASA is proposing to build its Lunar Gateway as a vehicle in a "Near-Rectilinear Halo Orbit" (NRHO) about the Moon. These halo orbits only exist in the presence of a third body, in this case, the Earth. The orbit is nearly orthogonal to the Moon's orbit about the Earth.

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As @DavidHammen already pointed out there is no restriction to two dimensions, the Lagrange points and the 2D planar orbits possible around some of them were "low hanging fruit" because they could be solved using pencil and paper or "writing with feathers using light from burning animal fat" so to speak.

Still, the stability of 3D three body orbits was at least suspected a long time ago but I don't yet know when the first halo orbit was proven to be stable.

For more on this, including some stunning visuals and videos, see

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