Painleve's transcendental differential equation PVI may be expressed as the consistency condition for a pair of linear differential equations with 2 by 2 matrix coefficients with rational entries. This linear system is assocaited with certain kernels which give trace class operatos on Hilbert space. This paper expresses such operators in terms of Hankel operators \Gamma_\phi of linear systems. For such, the Fredholm determinant \det (I-\Gamma_\phi P_(t, \infty )) gives rise to the tau function, which is shown to be the solution of a matrix Gelfand--Levitan equation. For meromorphic transfer functions that have poles on an arithmetric progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L^2(0, \infty ); so \det (I-\Gamma_\phi)can be expressed as a series of finite determinants. This applies to ellliptic functios of the second kind, such as satisfy Lame's differential equation.