Deformations of solutions of algebraic differential equations

As in algebraic geometry where the formal neighborhood of a point of a scheme contains informations about the singularities of the object, we extend this study to schemes where a point represents a solution of an algebraic differential equation. The obtained geometric object living naturally in a space of infinite dimension, a first step is to show that the formal neighborhood of a point not canceling the separant is noetherian, using considerations on the embedding dimension. We show that, in the neighborhood of points making the separant invertible, the embedding dimension is exactly the order of the considered differential equation. In a second step, we relate, for a certain type of differential equations of order two, the existence of essential singular components to the decrease of the embedding dimension, in the neighborhood of certain points.