  Einstein's equations with two commuting Killing vectors and the associated
Lax pair are considered. The equations for the connection $A(\varsigma, \eta,
\gamma)=\Psi_{,\gamma}\Psi^{-1}$, where $\gamma$ the variable spectral
parameter are considered. A transition matrix ${\cal T}= A(\varsigma, \eta,
\gamma)A^{-1}(\xi, \eta, \gamma)$ for $A$ is defined relating $A$ at ingoing
and outgoing light cones. It is shown that it satisfies equations familiar from
integrable pde's theory. A transition matrix on $\varsigma={\mbox constant}$ is
defined in an analogous manner.
  These transition matrices allow us to obtain a hierarchy of integrals of
motion with respect to time, purely in terms of the trace of a function of the
connections $g_{,\varsigma}g^{-1}$ and $g_{,\eta}g^{-1}$. Furthermore a
hierarchy of integrals of motion in terms of the curvature variable
$B=A_{,\gamma}A^{-1}$, involving the commutator $[A(1), A(-1)]$, is obtained.
  We interpret the inhomogeneous wave equation that governs $\sigma=ln N$, $N$
the lapse, as a Klein-Gordon equation, a dispersion relation relating energy
and momentum density, based on the first connection observable and hence this
first observable corresponds to mass. The corresponding quantum operators are
$\frac{\partial}{\partial t}$, $\frac{\partial}{\partial z}$ and this means
that the full Poincare group is at our disposal.
