  We revisit symplectic properties of the monodromy map for Fuchsian systems on
the Riemann sphere. We extend previous results of Hitchin, Alekseev-Malkin and
Korotkin-Samtleben where it was shown that the monodromy map is a Poisson
morphism between the Kirillov-Kostant Poisson structure on the space of
coefficients, on one side, and the Goldman bracket on the monodromy character
variety on the other. The extension is provided by defining larger spaces on
both sides which are equipped with symplectic structures naturally projecting
to the canonical ones. On the coefficient side our symplectic structure
corresponds to a non-degenerate quadratic Poisson structure expressed via the
rational dynamical $r$-matrix; it reduces to the Kirillov-Kostant bracket when
projected to the standard space. On the monodromy side we get a symplectic
structure which induces the symplectic structure of Alekseev-Malkin on the
leaves of the Goldman Poisson bracket. We prove that the monodromy map provides
a symplectomorphism using the formalism of Malgrange and one of the authors. As
a corollary we prove the recent conjecture by A.Its, O.Lisovyy and A.Prokhorov
in its "strong" version while the original "weak" version is derived from
previously known results. We show also that the isomonodromic Jimbo-Miwa
tau-function is intimately related to a generating function of such
transformation.
