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Let $Gr(d,n)$ be the Grassmannian of $d$-dimensional linear subspaces of an$n$-dimensional vector space $V$. A submanifold $X\subset Gr(d, n)$ gives riseto a differential system $\Sigma(X)$ that governs $d$-dimensional submanifoldsof $V$ whose Gaussian image is contained in $X$. We investigate a special caseof this construction where $X$ is a sixfold in $Gr(4, 6)$. The correspondingsystem $\Sigma(X)$ reduces to a pair of first-order PDEs for 2 functions of 4independent variables. Equations of this type arise in self-dual Ricci-flatgeometry. Our main result is a complete description of integrable systems$\Sigma(X)$. These naturally fall into two subclasses.  (1) Systems of Monge-Amp\`ere type. The corresponding sixfolds $X$ arecodimension 2 linear sections of the Pl\"ucker embedding$Gr(4,6)\subset\mathbb{P}^{14}$.  (2) General linearly degenerate systems. The corresponding sixfolds $X$ arethe images of quadratic maps $\mathbb{P}^6- Gr(4, 6)$ given by a version of theclassical construction of Chasles.  We prove that integrability is equivalent to the requirement that thecharacteristic variety of system $\Sigma(X)$ gives rise to a conformalstructure which is self-dual on every solution. In fact, all solutions carryhyper-Hermitian geometry.

Integrable systems in 4D associated with sixfolds in Gr(4,6)