The Geometry of Darboux Integrable Elliptic Systems

We characterize real elliptic differential systems whose solutions can be expressed in terms of holomorphic solutions to an associated holomorphic Pfaffian system $\mathcal{H}$ on a complex manifold. In particular, these elliptic systems arise as quotients by a group $G$ of the real differential system generated by the real and imaginary parts of $\mathcal{H}$, such that $G$ is the real form of a complex Lie group $K$ which is a symmetry group of $\mathcal{H}$. Subject to some mild genericity assumptions, we show that such elliptic systems are characterized by a property known as Darboux integrability. Examples discussed include first- and second-order elliptic PDE and PDE systems in the plane.