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