  Let U(n) be the unitary group, and $u(n)^*$ the dual of its Lie algebra,
equipped with the Kirillov Poisson structure. In their 1983 paper,
Guillemin-Sternberg introduced a densely defined Hamiltonian action of a torus
of dimension $(n-1)n/2$ on $u(n)^*$, with moment map given by the
Gelfand-Zeitlin coordinates. A few years later, Flaschka-Ratiu described a
similar, `multiplicative' Gelfand-Zeitlin system for the Poisson Lie group
$U(n)^*$.
  By the Ginzburg-Weinstein theorem, $U(n)^*$ is isomorphic to $u(n)^*$ as a
Poisson manifold. Flaschka-Ratiu conjectured that one can choose the
Ginzburg-Weinstein diffeomorphism in such a way that it intertwines the linear
and nonlinear Gelfand-Zeitlin systems. Our main result gives a proof of this
conjecture, and produces a canonical Ginzburg-Weinstein diffeomorphism.
