Integrable Lagrangians and modular forms

We investigate non-degenerate Lagrangians of the form ∫f(ux,uy,ut)dxdydt such that the corresponding Euler–Lagrange equations (fux)x+(fuy)y+(fut)t=0(fux)x+(fuy)y+(fut)t=0 are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density ff, are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the ‘master-Lagrangian’ corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball. We demonstrate how the knowledge of the symmetry group allows one to linearize the integrability conditions.