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The problem of $N$ particles interacting through pairwise central forces isnotoriously intractable for $N\geq3$. Some quite remarkable specific cases havebeen solved in one dimension, whereas higher-dimensional exactly solved systemsinvolve velocity-dependent or many-body forces. Here we show that the guidingcenter approximation---valid for charges moving in two dimensions in a strongconstant magnetic field---simplifies the three-body problem for an arbitraryinterparticle interaction invariant under rotations and translations and makesit solvable by quadratures. This includes a broad variety of special cases,such as that of three particles interacting through arbitrary pairwise centralpotentials. A spinorial representation for the system is introduced, whichallows a visualization of its phase space as the corresponding Bloch sphere aswell as the identification of a Berry-Hannay rotational anholonomy. Finally, abrief discussion of the quantization of the problem is presented.

The two-dimensional three-body problem in a strong magnetic field is integrable