Geodesic
Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential
Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 3, pp. 26-36

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We prove that if a complex Hamiltonian system with $n$ degrees of freedom has $n$ functionally independent meromorphic first integrals in involution and the monodromy group of the corresponding variational system along some phase curve has $n$ pairwise skew-orthogonal two-dimensional invariant subspaces, then the restriction of the action of this group to each of these subspaces has a rational first integral. The result thus obtained is applied to natural mechanical systems with homogeneous potential, in particular, to the $n$-body problem. 
@article{FAA_2000_34_3_a2,
     author = {S. L. Ziglin},
     title = {Integrals in {Involution} for {Groups} of {Linear} {Symplectic} {Transformations} and {Natural} {Mechanical} {Systems} with {Homogeneous} {Potential}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {26--36},
     publisher = {mathdoc},
     volume = {34},
     number = {3},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2000_34_3_a2/}
}
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S. L. Ziglin. Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential. Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 3, pp. 26-36. http://geodesic.mathdoc.fr/item/FAA_2000_34_3_a2/