Let M be an (n+1)-dimensional manifold and let a group G act transitively on M. I am going to make two assumptions about this action. First, I assume that there exists a hypersurface S₀ of M, such that the G-orbit of the (k-1)-st jet of S₀ in a point o of M is open in J^k-1(n,M) and, second, I assume that there are no open G-orbits in J^k(n,M). Then, starting from such an S₀, I will construct a family of (scalar) G-invariant k-th order PDEs in n independent variables and 1 dependent variable. I will show that the solutions to these equations are a natural analogue of the Weingarten surfaces in the three-dimensional Euclideam space. I will discuss in detail the cases when k=2 or k=3. I will also find convenient coordinates to locally describe the so-obtained equations.