Integrability of dispersionless differential equations in three and four dimensions: various approaches

First, I will tell on our work with David Calderbank in which we show that for general equations with quadratic characteristic manifold the existence of a Lax pair in vector fields is equivalent to the twistor approach. This explains why the maximal dimension for non-degenerate integrable equations may be 4. I'll shortly discuss what happens in higher dimensions.
Then, I will tell on a class of equations related to submanifolds of Grassmann g eometry, show the classification of integrable systems in this class, and discuss the difference between dimensions 3 and 4 in this context. Here the integrability is understood in the sense of the hydrodynamic reductions. This is a joint work with Boris Doubrov, Eugene Ferapontov, and Vladimir Novikov.