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Full symmetric Toda system and vector fields on the group $SO_n(\R)$
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abstract
In this paper we discuss the relation between the functions that give first integrals of full symmetric Toda system (an important Hamilton system on the space of traceless real symmetric matrices) and the vector fields on the group of orthogonal matrices: it is known that this system is equivalent to an ordinary differential equation on the orthogonal group, and we extend this observation further to its first integrals. As a by-product we describe a representation of the Lie algebra of $B^+(\R)$-invariant functions on the dual space of Lie algebra $\mathfrak{sl}_n(\R)$ (under the canonical Poisson structure) by vector fields on $SO_n(\R)$.
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The Lie -Bianchi integrability of the full symmetric Toda system
The paper identifies the Toda symmetry algebra with a central extension of the stochastic Lie algebra, but the claimed Lie-Bianchi integrability on the full phase space is not established.
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