  Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and
output space ${\mathbb C}^2$ and state space $H$. The scattering functions
$\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator
$\Gamma_{\phi_{(x)}}$; if $\Gamma_{\phi_{(x)}}$ is trace class, then the
Fredholm determinant $\tau (x)=\det (I+\Gamma_{\phi_{(x)}})$ determines the tau
function of $(-A,B,C)$. The paper establishes properties of algebras including
$R_x=\int_x^\infty e^{-tA}BCe^{-tA}dt$ on $H$. Thus the paper obtains solutions
of the sinh-Gordon PDE. The tau function for sinh-Gordon satisfies a particular
Painl\'eve $\mathrm{III}'$ nonlinear ODE and describes a random matrix model,
with asymptotic distribution found by the Coulomb fluid method to be the
solution of an electrostatic variational problem on an interval.
