The space $M(n)$ of $n\times n$ matrices is a Poisson manifold. Gelfand-Zeitlin theory gives rise a maximal Poisson commutative algebra of functions on $M(n)$. We show that the corresponding Poisson vector fields are globally integrable and give to a new commutative group $A$ of Poisson automorphisms on M(n). The orbits of $A$ are explicitly given and give rise to new decompositions of $M(n)$. \vskip .1in \noindent The group $A$ leads to a solution of a classical analogue of the Gelfand-Kirillov conjecture