Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian

Familiar examples include the Boyer-Finley equation uxx+uyy = eutt , the potential form of the dispersionless Kadomtsev-Petviashvili (dKP) equation uxt−1 2u2 xx = uyy, the dispersionless Hirota equation ( − )euxy + ( − )euyt + ( − )eutx = 0, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit.