I am trying to understand the article Reduction of Singularities of the Differential Equation $Ady=Bdx$ by Arno van den Essen.

Let $A,B\in k[[X,Y]]$ be formal power series and $D$ the vector field $A\frac{\partial}{\partial X}+B\frac{\partial}{\partial Y}$. In the paper, a solution of $D$ at $(0,0)$ is defined as a nonzero nonunit formal power series $F\in k[[X,Y]]$ such that $DF \equiv 0 \mod F $ in $k[[X,Y]]$.

My aim is to describe explicitly all the solutions of $D$ at $(0,0)$ for some particular values of $A$ and $B$, let's say $A=X(Y+X^2)$ and $B=Y^2(1+Y)$ for instance. Are there any results in the litterature to tackle this problem? And in the general case, is it possible to describe the solution set? These questions weren't adressed in the aforementioned paper (unless I didn't understand it correctly).