  We investigate non-degenerate Lagrangians of the form
  $$
  \int f(u_x, u_y, u_t) dx dy dt
  $$ such that the corresponding Euler-Lagrange equations
  $
  (f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_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 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. We demonstrate
how the knowledge of the symmetry group allows one to linearise the
integrability conditions.
