I have a system of equations that can be put in the form,
$\begin{align}\nabla\cdot \mathbf{n}=\mathbf{T}\cdot \mathbf{n}\end{align}\\\nabla\times \mathbf{n}=\mathbf{T}\times \mathbf{n},$
with $\mathbf{T}$ a vector such that $\nabla\cdot \mathbf{T}=0,\: \mathbf{T}=(f(y),g(x),0)$, and $\mathbf{n}$ has to lie on the plane, that is, $\mathbf{n}=(n_x(x,y),n_y(x,y),0)$. Also some regular data on a closed curve on the plane can be provided. The question is how to find $\mathbf{n}$ ? In terms of $\mathbf{T}$, course.
The system is well posed, so it has solution, but so far that's all I can say. I'd like to have a explicit integral. The most similar problem that I've seen is that of $\nabla\times f=f$, but here I'm much more constrained.
Thanks in advance,