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We study higher-order conservation laws of the non-linearizable ellipticPoisson equation $ \frac{{\partial}^2 u}{\partial z \partial \bar{z}} = -f(u) $as elements of the characteristic cohomology of the associated exteriordifferential system. The theory of characteristic cohomology determines anormal form for differentiated conservation laws by realizing them as elementsof the kernel of a linear differential operator. The S^1--symmetry of the PDEleads to a normal form for the undifferentiated conservation law as well. Weshow 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 realtwo--dimensional space of new conservation laws in normal form appears at eacheven prolongation. When $f_{uu}=\beta f$ this upper bound is attained and thework of Pinkall and Sterling allows them to be written explicitly. We relatehigher-order conservation laws to generalized symmetries of the exteriordifferential system by identifying their generating functions. This Noethercorrespondence provides the connection between conservation laws and thecanonical Jacobi fields of Pinkall and Sterling.

Higher-order conservation laws for the non-linear Poisson equation via the characteristic cohomology