  We show that for $U(1)$-symmetric spacetimes on $S^3 \times R$ a constant of
motion associated with the well known Geroch transformation, a functional
$K[h_{ij},\pi^{ij}]$, quadratic in gravitational momenta, is strictly positive
in an open subset of the set of all $U(1)$-symmetric initial data, and
therefore not weakly zero. The Mixmaster initial data appear to be on the
boundary of that set. We calculate the constant of motion perturbatively for
the Mixmaster spacetime and find it to be proportional to the minisuperspace
Hamiltonian to the first order in the Misner anisotropy variables, i.e. weakly
zero. Assuming that $K$ is exactly zero for the Mixmaster spacetime, we show
that Geroch's transformation, when applied to the Mixmaster spacetime, gives a
new \mbox{$U(1)$-symmetric} solution of the vacuum Einstein equations, globally
defined on \mbox{$S^2 \times S^1 \times R$},which is non-homogeneous and
presumably exhibits Mixmaster-like complicated dynamical behavior.
