  In [Alekseevsky, Gutt, Manno, Moreno: "A general method to construct
invariant PDEs on homogeneous manifolds", Communications in Contemporary
Mathematics (2021)] the authors have developed a method for constructing
$G$-invariant PDEs imposed on hypersurfaces of an $(n+1)$-dimensional
homogeneous space $G/H$, under mild assumptions on the Lie groups $G$. In the
present paper the method is applied to the case when $G=\mathsf{PGL}(n+1)$ or
$G=\mathsf{Aff}(n+1)$ and the homogeneous space $G/H$ is the
$(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ or affine $\mathbb{A}^{n+1}$
space, respectively. The paper's main result is that projectively or affinely
invariant PDEs with $n$ independent and one unknown variables are in one-to-one
correspondence with $\mathsf{CO}(d,n-d)$-invariant hypersurfaces of the space
of trace-free cubic forms in $n$ variables. Local descriptions are also
provided.
