Alexander Odesskii (University of Manchester)
We investigate non-degenerate Lagrangians giving the actions of the form
∫ f(ux, uy, ut) dx dy dt
such that the corresponding Euler-Lagrange equations
(fux)x+ (fuy)y+ (fut)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.