The Fredholm determinants of a special class of integral operators K
supported on the union of m curve segments in the complex plane are shown to be
the tau-functions of an isomonodromic family of meromorphic covariant
derivative operators D_l. These have regular singular points at the 2m
endpoints of the curve segments and a singular point of Poincare index 1 at
infinity. The rank r of the vector bundle over the Riemann sphere on which they
act equals the number of distinct terms in the exponential sums entering in the
numerator of the integral kernels. The deformation equations may be viewed as
nonautonomous Hamiltonian systems on an auxiliary symplectic vector space M,
whose Poisson quotient, under a parametric family of Hamiltonian group actions,
is identified with a Poisson submanifold of the loop algebra Lgl_R(r) with
respect to the rational R-matrix structure. The matrix Riemann-Hilbert problem
method is used to identify the auxiliary space M with the data defining the
integral kernel of the resolvent operator at the endpoints of the curve
segments. A second associated isomonodromic family of covariant derivative
operators D_z is derived, having rank n=2m, and r finite regular singular
points at the values of the exponents defining the kernel of K. This family is
similarly embedded into the algebra Lgl_R(n) through a dual parametric family
of Poisson quotients of M. The operators D_z are shown to be analogously
associated to the integral operator obtained from K through a Fourier-Laplace
transform.