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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).

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    $\begingroup$ I am just curious, what is the reason they add $\text{mod }F$ there? For example, in the case $A=1$, $B=0$ I would expect the solution to satisfy $F_x =0$ but instead we get $F_x$ is proportional to $F.$ $\endgroup$
    – Ilia
    Commented Feb 16, 2022 at 16:34
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    $\begingroup$ Hello, the author of the paper shows under certain assumptions that starting from solutions of $Ady=Bdx$, you can obtain solutions of $D=A\frac{\partial}{\partial X}+B\frac{\partial}{\partial Y}$ at $(0,0)$ (and vice versa). It just happens that when you write the proof, you don't find an $F$ such that $A\frac{\partial F}{\partial X}+B\frac{\partial F}{\partial Y}$ is exactly zero, it is only a multiple of $F$. $\endgroup$
    – b.b
    Commented Feb 16, 2022 at 20:40

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