A general method to construct invariant PDEs on homogeneous manifolds

Let M=G/H be an (n+1)-dimensional homogeneous manifold and Jk(n,M)=:Jk be the manifold of k-jets of hypersurfaces of M. The Lie group G acts naturally on each Jk. A G-invariant partial differential equation of order k for hypersurfaces of M (i.e., with n independent variables and 1 dependent one) is defined as a G-invariant hypersurface E⊂Jk. We describe a general method for constructing such invariant partial differential equations for k≥2. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup H(k−1) of the (k−1)-prolonged action of G. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space En+1 and in the conformal space Sn+1. Our method works under some mild assumptions on the action of G, namely: A1) the group G must have an open orbit in Jk−1, and A2) the stabilizer H(k−1)⊂G of the fiber Jk→Jk−1 must factorize via the group of translations of the fiber itself.