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As in the case of Mumford's geometric invariant theory (which concerns projective good quotients) the problem can be reduced to the case of an action of a torus.
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We investigate the eigenvalue problem for such systems and the correspondingD-module when the eigenvalues are in generic position.
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In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues.
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This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues.
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A counterexample to Hilbert's Fourteenth Problem in dimension six
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