Is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities?

This is in connection with a quote from someone on the web that I saw a long time ago. At that time I had contacted the author, but they chose not to answer.

The quote:

In some sense, the space of all possible linear PDE's can be viewed as a singular algebraic variety, where Hormander's theory applies only to generic (smooth) points and the most interesting and heavily studied PDE's all lie in a lower-dimensional subvariety and mostly in the singular set of the variety.

Any pointers/refs on any of the points made in the quote would be gratefully received...