Let $A$ be the algebra of all smooth functions $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is flat at the origin and is real analytic on $\mathbb{R}^2 \setminus \{0\}$.

Let $B $ be the subalgebra of $A$ consisting of all Schwartz functions in $A$.

For every polynomial vector field $X = P(x,y) \partial_{x} + Q(x,y) \partial_{y}$ we define the differential operator $D_{X}(U)=PU_{x}+QU_{y}$. Obviously $A$ and $B$ are invariant under this differential operator.

Let $X=(y-(x^3 -x))\partial_{x}-x \partial_{y}$ be the Van der Pol vector field.

What can be said about the codimension of the range of $D_{X}:A \to A$? Is it finite?

What can be said about the codimension of the range of $D_{X}: B \to B$? Is it finite?

We explain about the motivation for consideration of such $A$ and $B$:

We require the real analyticity on the punctured plane to avoid the obvious infinite codimension since if a limit cycle surrounds a non resonance singularity, using bump functions, one can show that the codimension is infinity, as we explained here. We require the flatness at the unique singularity at the origin to avoid some obstruction for existence of (even) formal power series solutions to the equation $D_{X}.g=f$ when $X$ is the Van der Pol equation or a more general algebraic vector field with degenerate singularity (vanishing some first Jets at the origin). The other reason for this flat requirement is that we would like to not engage with the problem of global analytic extension of a local real analytic solution (if it exists) to $D_{X}.g=f$. For definition of $B$, we require the Schwartz condition in order to apply the Fourier transform to convert a first order PDE, associated with a quadratic system, to higher order PDE to have a possible chance to work with an elliptic PDE. This is explained in the Remark 2 and its consecutive example of page 5 of the following note:

https://arxiv.org/pdf/1302.0001.pdf

Finally, for the Van der Pol vector field $X$, what can be said about the codimension of the range of $D_{X}$ as an operator on the space $C^{\omega}(\mathbb{R}^2)$, the space of real analytic functions on the plane?

What about the codimension of the range of the operator $L_X$ when it acts on either $\Omega^1(\mathbb{R}^2)$ or $\Omega^1(\mathbb{R}^2)/Z^1(\mathbb{R}^2)$ where $Z^1(\mathbb{R}^2)$ is the space of closed $1$-forms? Do we have a possible chance for "finite codimension" in the latter quotient operator $L_X:\Omega^1(\mathbb{R}^2)/(Z^1(\mathbb{R}^2) \to \Omega^1(\mathbb{R}^2)/(Z^1(\mathbb{R}^2)$?

Note that the codimension of the range of $L_X$ is an upper bound for the number of closed orbits of $X$ The reason is written in the motivation part of the following post: Integral Separation of disjoint submanifolds of $\mathbb{R}^n$