We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampere property in any dimension $\geq 4$. 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-Ampere 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.

中文翻译：

---

无色散 Hirota 型方程的可积性和辛 Monge-Ampère 性质

我们证明了无色散 Hirota 型方程的可积性意味着在任何维度 $\geq 4$ 中的辛 Monge-Ampere 性质。在 4D 中，这通过一系列在自双引力中产生的天体方程产生了 Hirota 型可积无色散偏微分方程的完整分类。作为我们方法的副产品，我们推导出了一个对任何维度的辛 Monge-Ampere 方程进行表征的对合关系系统。此外，我们证明了在 4D 中，可积性的要求等效于由方程在每个解上的特征变化定义的共形结构的自对偶性，这又等效于无色散 Lax 对的存在。我们还通过相应共形结构的平坦度给出了 Hirota 型方程的线性化标准，