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Einstein's equations with two commuting Killing vectors and the associatedLax pair are considered. The equations for the connection $A(\varsigma, \eta,\gamma)=\Psi_{,\gamma}\Psi^{-1}$, where $\gamma$ the variable spectralparameter are considered. A transition matrix ${\cal T}= A(\varsigma, \eta,\gamma)A^{-1}(\xi, \eta, \gamma)$ for $A$ is defined relating $A$ at ingoingand outgoing light cones. It is shown that it satisfies equations familiar fromintegrable pde's theory. A transition matrix on $\varsigma={\mbox constant}$ isdefined in an analogous manner.  These transition matrices allow us to obtain a hierarchy of integrals ofmotion with respect to time, purely in terms of the trace of a function of theconnections $g_{,\varsigma}g^{-1}$ and $g_{,\eta}g^{-1}$. Furthermore ahierarchy 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 energyand momentum density, based on the first connection observable and hence thisfirst observable corresponds to mass. The corresponding quantum operators are$\frac{\partial}{\partial t}$, $\frac{\partial}{\partial z}$ and this meansthat the full Poincare group is at our disposal.

Observables in terms of connection and curvature variables for Einstein's equations with two commuting Killing vectors