  The level spacing distributions in the Gaussian Unitary Ensemble, both in the
``bulk of the spectrum,'' given by the Fredholm determinant of the operator
with the sine kernel ${\sin \pi(x-y) \over \pi(x-y)}$ and on the ``edge of the
spectrum,'' given by the Airy kernel ${\rm{Ai}(x) \rm{Ai}'(y) - \rm{Ai}(y)
\rm{Ai}'(x) \over (x-y)}$, are determined by compatible systems of
nonautonomous Hamiltonian equations. These may be viewed as special cases of
isomonodromic deformation equations for first order $ 2\times 2 $ matrix
differential operators with regular singularities at finite points and
irregular ones of Riemann index 1 or 2 at $\infty$. Their Hamiltonian structure
is explained within the classical R-matrix framework as the equations induced
by spectral invariants on the loop algebra ${\tilde{sl}(2)}$, restricted to a
Poisson subspace of its dual space ${\tilde{sl}^*_R(2)}$, consisting of
elements that are rational in the loop parameter.
