Tracy and Widom showed that fundamentally important kernels in random matrix theory arise from systems of differential equations with rational coefficient. More generally, this paper considers symmetric Hamiltonian systems and determines the properties of kernels that arise from them. The inverse spectral problem for self-adjoint Hankel operators gives sufficient condition for a self-adjoint operator to be the Hankel operator on L^2(0, \infinity ) from a linear system in continuous time; so this paper expresses certain kernels as squares of Hankel operators. For suitable systems (-A,B,C) with one dimensional input and output spaces, there exists a Hankel operator \Gamma with kernel \phi such that \det (I+(z-1)\Gamma\Gamma^\dagger) is the generating function of a dterminnatal random point field on (0, \infty ). The invrse scattering transform for the Zhakarov--Shabat system involves an Gelfand--Levitan integral equation such that the daigonal of te solution gives the derivative of th log generating function. Some dtererminatal print fields in random matrix theory satisfy similar results.