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We investigate dispersionless integrable systems in 3D associated withfourfolds in the Grassmannian Gr(3,5). Such systems appear in numerousapplications in continuum mechanics, general relativity and differentialgeometry, and include such well-known examples as the dispersionlessKadomtsev-Petviashvili equation, the Boyer-Finley equation, etc. We prove theequivalence of the four different approaches to integrability, revealing aremarkable correspondence with Einstein-Weyl geometry and the theory of GL(2,R)structures.

On the integrability in Grassmann geometries: integrable systems associated with fourfolds Gr(3, 5)