  We characterize 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. The integrability conditions constitute an
over-determined system of fourth order PDEs for the Lagrangian density $f$,
which is in involution and possess interesting differential-geometric
properties. The moduli space of integrable Lagrangians, factorized by the
action of a natural equivalence group, is three-dimensional. Familiar examples
include the dispersionless Kadomtsev-Petviashvili (dKP) and the Boyer-Finley
Lagrangians, $f=u_x^3/3+u_y^2-u_xu_t$ and $f=u_x^2+u_y^2-2e^{u_t}$,
respectively. A complete description of integrable cubic and quartic
Lagrangians is obtained. Up to the equivalence transformations, the list of
integrable cubic Lagrangians reduces to three examples, $ f=u_xu_yu_t,
f=u_x^2u_y+u_yu_t, and f=u_x^3/3+u_y^2-u_xu_t ({\rm dKP}).$ There exists a
unique integrable quartic Lagrangian, $ f=u_x^4+2u_x^2u_t-u_xu_y-u_t^2.$ We
conjecture that these examples exhaust the list of integrable polynomial
Lagrangians which are essentially three-dimensional (it was verified that there
exist no polynomial integrable Lagrangians of degree five). We prove that the
Euler-Lagrange equations are integrable by the method of hydrodynamic
reductions if and only if they possess a scalar pseudopotential playing the
role of a dispersionless `Lax pair'. MSC: 35Q58, 37K05, 37K10, 37K25. Keywords:
Multi-dimensional Dispersionless Integrable Systems, Hydrodynamic Reductions,
Pseudopotentials.
