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Familiar examples include the Boyer-Finley equation uxx+uyy = eutt , the potential
form of the dispersionless Kadomtsev-Petviashvili (dKP) equation uxt−1
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.