Projectively and affinely invariant PDEs on hypersurfaces

In Communications in Contemporary Mathematics 24 3, (2022), the authors have developed a method for constructing G-invariant partial differential equations (PDEs) imposed on hypersurfaces of an (n + 1)-dimensional homogeneous space G=H, under mild assumptions on the Lie group G. In the present paper, the method is applied to the case when G = PGL(n + 1) (respectively, G = Aff(n + 1)) and the homogeneous space G=H is the (n + 1)-dimensional projective P^{n+1} (respectively, affine A^{n+1}) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with n independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms in n variables with respect to the group CO(d,n-d) of conformal transformations of R^{d,n-d}.