Einstein’s equations with two commuting Killing vectors and the associated Lax pair are considered. The equations for the connection
A
(
ς
,
η
,
γ
)=
Ψ
,
γ
Ψ
−1
, where
γ
the variable spectral parameter are considered. A transition matrix
T
=
A
(
ς
,
η
,
γ
)
A
−1
(
ξ
,
η
,
γ
) for
A
is defined relating
A
at ingoing and outgoing light cones. It is shown that it satisfies equations familiar from integrable PDE theory. A transition matrix on
ς
= 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
,
ς
g
−1
and
g
,
η
g
−1
. Furthermore, a hierarchy of integrals of motion in terms of the curvature variable
B
=
A
,
γ
A
−1
, involving the commutator [
A
(1),
A
(−1)], is obtained. We interpret the inhomogeneous wave equation that governs
σ
=
lnN
,
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 ∂/∂
t
, ∂/∂
z
and this means that the full Poincare group is at our disposal.
Exact solutions of Einstein's equations with ideal gas sources
