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 $\partial_x+\partial_y, \partial_x+\partial_y$.

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 $\partial_x+\partial_y, \partial_x-\partial_y$.