I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a potential function (which was addressed in a separate [question](https://mathoverflow.net/questions/122308/when-a-riemannian-manifold-is-of-hessian-typ)). In that question, Robert Bryant noted that the characteristic variety consists of three points. Concretely, I'm wondering if this means that there are at most three solutions to this system of equations (perhaps modulo some affine transformations).


More generally, does the number of points in the characteristic variety bound the number of real analytic solutions for a given system of PDEs?


My understanding is that the characteristic variety gives something like the formal power series solutions to the system, so there shouldn't be more real analytic solutions than that. However, my knowledge of $D$-modules is really lacking, so I was wondering if anyone could either correct me on this or else point me to a good reference to learn more. I've been trying to read through Bryant's Exterior Differential Systems, and I apologize if this question is obvious for those who understand the theory.



Can the number of solutions to a system of PDEs be bounded using the characteristic variety?
The wave equation in the plane is $\partial^2\_x-\partial^2\_y=(\partial\_x+\partial\_y)(\partial\_x-\partial\_y)$, so two points in the characteristic variety, but infinite dimensional family of solutions.


Each isolated point in the characteristic variety represents a hypersurface foliation inside every sufficiently smooth solution, by a theorem of Ofer Gabber. For the wave equation, this is the foliation by the two directions spanned by the vector fields $\partial\_x+\partial\_y, \partial\_x-\partial\_y$.


