By A T Fomenko; V V Trofimov

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**Additional resources for Integrable systems on Lie algebras and symmetric spaces**

**Example text**

E. that Ker w c Hx. Let w(X, Y ) = 0 for an y vector y E �M;. The vector X m u st be represented in the form X = s grad h for a given function h E H�. 1 to denote the orthogonal complem ent to pl ane TxM ; in Tx M relative to th e fonn w. Since the form is non-d egenerate , the equality dim(T� M� )- = dim M - dim � M � = dim V = k is valid. We should bear in mind that in the case of skew-symmetric scalar p rod ucts the space Tx M need not necessarily decompose into the direct sum of � M � and ( TxM �)� , since t hese planes can have a non-zero intersection.

The arb i trary basis H I ' . . in (over C) generates the basis { H� } in Ho (over C) and in R (over C) . T h is basis can be supplemented by vecto rs E" E G\ a i= 0, a E A. Vectors Ez may be chosen so that B(E", E - z) = 1 . The commutation operation in G is then written as follows : Ho Ho); Ho. (Hk+d J1(Hk+d. p EIl+/I' (J B(h, H�) = a(h) , - H�; :x + f3 #- 0 root a + f3 #- 0 non-root h e R. The vectors £:1. E G % may be chosen so that N%(J = N - 2 - P The con s ta nt s N z/J m ay be taken to be real (after the approp ri a te normalization of ' A.

Fig. 14. e. that all orbits of the group (f) close to an orbit (ij(x) are difTeomorphic to it. Let us examine the projection p : -+ M/(f> of manifold M on the orbit space M/