Consider the following PDE: \begin{equation} p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$} \end{equation} where $g$ is a flat function at the point (0,0).

Let $X$ denote the the vector field $p \frac{\partial}{\partial p}-q \frac{\partial}{\partial q}$. The equation $\star$ can be also written as $\mathcal{L}_Xf=g,$ where $\mathcal{L}_Xf$ stands for the Lie derivative of $f$ along the vector field $X.$

I need to prove that there exists at least one $C^\infty$ smooth solution to the equation $\star$ in some neighbourhood of the point $(0,0).$ I also want this solution to be flat at the point $(0,0).$

The method of characteristics doesn't really work in this situation since (0,0) is a singular point of the vector field $X.$ The characteristics of $\star$ are given by the level sets of the function $pq.$ So it is easy to find a smooth solution in the domain $\mathbb R^2 \setminus \{(p, q)|pq=0\}.$ For example, the initial data can be given as: $f(p, q)=0$ if $p=q$ or $p=-q$.

But it is unclear whether or not this solution can be extended to some neighbourhood of the point $(0,0).$

I was thinking about this problem for quite a while but with no success so far.