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Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V n . A submanifold X ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V n whose Gaussian image is contained in X. Systems of the form Σ(X) appear in numerous applications in continuum mechanics, theory of integrable systems, general relativity and differential geometry. They include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, Plebańsky's heavenly equations and so on.In this paper we concentrate on the particularly interesting case of this construction where X is a fourfold in Gr(3, 5). Our main goal is to investigate differential-geometric and integrability aspects of the corresponding systems Σ(X). We demonstrate the equivalence of several approaches to dispersionless integrability such as• the method of hydrodynamic reductions;• the method of dispersionless Lax pairs; • integrability on solutions, based on the requirement that the characteristic variety of system Σ(X) defines an Einstein-Weyl geometry on every solution; • integrability on equation, meaning integrability (in twistor-theoretic sense) of the canonical GL(2, R) structure induced on a fourfold X ⊂ Gr(3, 5).All these seemingly different approaches lead to one and the same class of integrable systems Σ(X). We prove that the moduli space of such systems is six-dimensional. We give a complete description of linearisable systems (the corresponding fourfold X is a linear section of Gr(3, 5)) and linearly degenerate systems (the corresponding fourfold X is the image of a quadratic map P 4 Gr(3, 5)). The fourfolds corresponding to 'generic' integrable systems are not algebraic, and can be parametrised by generalised hypergeometric functions.

We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampère property in any dimension ≥ 4. In 4D this yields a complete classification of integrable dispersionless PDEs of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-product of our approach we derive an involutive system of relations characterising symplectic Monge-Ampère equations in any dimension.Moreover, we demonstrate that in 4D the requirement of integrability is equivalent to self-duality of the conformal structure defined by the characteristic variety of the equation on every solution, which is in turn equivalent to the existence of a dispersionless Lax pair. We also give a criterion of linerisability of a Hirota type equation via flatness of the corresponding conformal structure, and study symmetry properties of integrable equations. MSC: 35L70, 35Q51, 35Q75, 53A30, 53Z05.

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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 ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this construction where X is a six-fold in Gr(4, 6). The corresponding system Σ(X) reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems Σ(X). These naturally fall into two subclasses. Systems of Monge–Ampère type. The corresponding six-folds X are codimension 2 linear sections of the Plücker embedding Gr(4, 6)$ \hookrightarrow \mathbb{P}^{14}$. General linearly degenerate systems. The corresponding six-folds X are the images of quadratic maps $\mathbb{P}^{6}\dashrightarrow \ $Gr(4, 6) given by a version of the classical construction of Chasles. We prove that integrability is equivalent to the requirement that the characteristic variety of system Σ(X) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.