A new approach to the construction of isomonodromy deformations of 2 x 2 Fu
chsian systems is presented. The method is based on a combination of the al
gebrogeometric scheme and Riemann-Hilbert approach of the theory of integra
ble systems. For a given number 2g + 1, g greater than or equal to 1, of fi
nite (regular) singularities, the method produces a 2g-parameter submanifol
d of the Fuchsian monodromy data for which the relevant Riemann-Hilbert pro
blem can be solved in closed form via the Baker-Akhiezer function technique
. This in turn leads to a 2g-parameter family of solutions of the correspon
ding Schlesinger equations, explicitly described in terms of Riemann theta
functions of genus g. In the case g = 1 the solution found coincides with t
he general elliptic solution of the particular case of the Painleve VI equa
tion first obtained by N. J. Hitchin [H1].