  The classical scattering of cylindrical gravitational waves is exactly
solvable. The motivation for this paper is to understand if the quantum
scattering problem is also exactly solvable. The classical dynamics is governed
by a two dimensional sigma model. We study this sigma model's $S$-matrix. We
construct a conserved nonlocal charge and derive the associated tree-level
$S$-matrix conservation law. We check our conservation law directly using
Feynman diagrams. The existence of this symmetry is a hint that cylindrical
gravitational waves might have an exactly solvable $S$-matrix.
