Manchester Geometry Seminar 2007/2008

We investigate non-degenerate Lagrangians giving the actions of the form

∫ f(u_x, u_y, u_t) dx dy dt

such that the corresponding Euler-Lagrange equations

(f_ux)_x+ (f_uy)_y+ (f_ut)_t=0

are integrable by the method of hydrodynamic reductions. The integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f, 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.