Geodesic
The symplectic structure of the orbits of the coadjoint representation of Lie algebras of type $E\underset{\rho}\times G$
Sbornik. Mathematics, Tome 51 (1985) no. 1, pp. 275-286 
T. A. Pevtsova. The symplectic structure of the orbits of the coadjoint representation of Lie algebras of type $E\underset{\rho}\times G$. Sbornik. Mathematics, Tome 51 (1985) no. 1, pp. 275-286. http://geodesic.mathdoc.fr/item/SM_1985_51_1_a17/
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     title = {The symplectic structure of the orbits of the coadjoint representation of {Lie} algebras of type $E\underset{\rho}\times G$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The following theorem is proved. Theorem. Let $G$ be the semidirect sum of a simple Lie algebra $H$ and an Abelian algebra relative to representation $\mu$. Then a complete involutive system of rational functions on $G^*$ is explicitly constructed in the following cases: a) {\it$H=\operatorname{gl}(2n)$ and $\mu=\Lambda^2\rho$;} b) {\it$H=\operatorname{sl}(2n)$ and $\mu=s^2\rho$;} c) {\it$H=\operatorname{sp}(2n)$ and $\mu=\rho+\tau$, where $\rho$ is the minimal representation and $\tau$ is the one-dimensional trivial representation.} Bibliography: 9 titles.

[1] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[2] Mischenko A. S., Fomenko A. T., “Uravneniya Eilera na konechnomernykh gruppakh Li”, Izv. AN SSSR. Seriya matem., 42:2 (1978), 396–415 | MR | Zbl

[3] Trofimov V. V., “Uravneniya Eilera na borelevskikh podalgebrakh poluprostykh algebr Li”, Izv. AN SSSR. Seriya matem., 43 (1979), 714–732 | MR | Zbl

[4] Trofimov V. V., “Uravneniya Eilera na konechnomernykh razreshimykh gruppakh Li”, Izv. AN SSSR. Seriya matem., 44 (1980), 1191–1200 | MR

[5] Trofimov V. V., Fomenko A. T., “Metodika postroeniya gamiltonovykh potokov na simmetricheskikh prostranstvakh i integriruemost nekotorykh gidrodinamicheskikh sistem”, DAN SSSR, 254:6 (1980), 1349–1353 | MR

[6] Pevtsova T. A., “Odin sposob postroeniya kommutativnoi algebry integralov na algebrakh Li”, Sedmaya konferentsiya po sovremennym problemam geometrii (tezisy dokl.), Minsk, 1974, 149

[7] Rais M., “L'indice des produits semi-directs $E\underset{\rho}{\times}G$”, C. r. Acad. Sci. Paris, 287 (1978), 195–197 | MR | Zbl

[8] Andreev E. M., Vinberg E. B., Elashvili A. G., Funkts. analiz, 1:4 (1967), 3–7 | MR | Zbl

[9] Vergne M., “La structure de Poisson sur 1 algebre summetrigue d'une algebre de Lienilpotente”, Bull. Soc. Math. France, 100 (1972), 301–335 | MR | Zbl