It is an easy exercise to show that the two-dimensional Monge-Ampère equations are the only two-dimensional second-order PDEs that are invariant under the natural action of the affine group of the plane. In three dimensions, an analogous statement can be proved, though it requires much more computations. In four dimensions, computations are simply unendurable, and the necessity of a more conceptual approach to the problem begins to show. In this talk I will recall that hypersurfaces of Lagrangian Grassmannians and second-order PDEs are basically the same thing, so that the notion of the invariancy (with respect to a given Lie group G) of a (multidimensional) second-order PDE can be formulated in terms of the G-invariancy of the corresponding hypersurface of the Lagrangian Grassmannian. Via the Plucker embedding, hypersurfaces of Lagrangian Grassmannians can be embbeded in a projective space. Such a projective space turns out to be a natural G-module, so that repesentation theory can be used for finding all the (relative) G-invariants polynomials, whose zero loci corresponds to G-invariant hypersurfaces. Up to 3 independent variables, such a method reveals nothing new, and it is just another way to show that Monge-Ampère equations correspond precisely to GL(n)-invariant hypersurfaces. Surprisingly enough, for n=4, a new unexpected class of invariant second-order PDEs pops out, which is not made of Monge-Ampère equations. This talk is based on a joint work with D. Alekseevsky and G. Manno.