Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian

We investigate integrable second-order equations of the form $F(u_{xx},u_{xy},u_{yy},u_{xt},u_{yt},u_{tt})=0,$ which typically arise as the Hirota-type relations for various (2 + 1)-dimensional dispersionless hierarchies. Familiar examples include the Boyer–Finley equation $u_{xx}+u_{yy}=e^{u_{tt}}$, the potential form of the dispersionless Kadomtsev–Petviashvili (dKP) equation $u_{xt}\frac {1}{2}u^2_{xx}=u_{yy}$, the dispersionless Hirota equation $(\alpha-\beta)e^{u_{xy}}+(\beta-\gamma)e^{u_{yt}}+(\gamma-\alpha)e^{u_{tx}}=0$, etc. The integrability is understood as the existence of an infinity of 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.