  We study higher-order conservation laws of the non-linearizable elliptic
Poisson equation $ \frac{{\partial}^2 u}{\partial z \partial \bar{z}} = -f(u) $
as elements of the characteristic cohomology of the associated exterior
differential system. The theory of characteristic cohomology determines a
normal form for differentiated conservation laws by realizing them as elements
of the kernel of a linear differential operator. The S^1--symmetry of the PDE
leads to a normal form for the undifferentiated conservation law as well. We
show that for higher-order conservation laws to exist, it is necessary that $f$
satisfies a linear second order ODE. In this case, an at most real
two--dimensional space of new conservation laws in normal form appears at each
even prolongation. When $f_{uu}=\beta f$ this upper bound is attained and the
work of Pinkall and Sterling allows them to be written explicitly. We relate
higher-order conservation laws to generalized symmetries of the exterior
differential system by identifying their generating functions. This Noether
correspondence provides the connection between conservation laws and the
canonical Jacobi fields of Pinkall and Sterling.
