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In recent work Bertram Kostant and Nolan Wallach ([KW1], [KW2]) have definedan interesting action of a simply connected Lie group $A$ isomorphic to\mathbb{C}^{{n\choose 2}} on gl(n) using a completely integrable system derivedfrom Gelfand-Zeitlin theory. In this paper we show that an analogous action of\mathbb{C}^{d} exists on the complex orthogonal Lie algebra so(n), where d ishalf the dimension of a regular adjoint orbit in so(n). In [KW1], Kostant andWallach describe the orbits of $A$ on a certain Zariski open subset of regularsemisimple elements in gl(n). We extend these results to the case of so(n). Wealso make brief mention of the author's results in [Col1], which describe all$A$-orbits of dimension {n\choose 2} in gl(n).

The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n)