Coordinates Free Cauchy-Riemann equations Cauchy-Riemann equations can be seen as a system of 2 PDEs for two functions on the plane:
$$
\begin{align*}
L_{\frac{\partial}{\partial x}}(u) &= L_{\frac{\partial}{\partial y}}(v) \\ 
L_{\frac{\partial}{\partial y}}(u) &= -L_{\frac{\partial}{\partial x}}(v)
\end{align*}
$$
where $L_X$ denotes the Lie derivative along vector field $X$. Since the vector fields $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ commutes, we have natural integrability conditions:
$$
\begin{align*}
\Delta u &= 0\\ 
\Delta v &= 0,
\end{align*}
$$
and we know that there exists smooth (even analytic) solutions.
I am surprised that I did not found any results about the natural generalisation of those equations, namely:
$$
\begin{align*}
L_A(u)&=L_B(v)\\
L_B(u)&=-L_A(v)
\end{align*}
$$
where $A$ and $B$ are any linearly independant smooth vector fields on $\mathbb{R}^2$. We can also derive integrability conditions but we obtain two complicate elliptic PDEs and it is not trivial that a solution exists.
Question: Is there references treating this equations? and giving conditions for the existence of $C^{\infty}$ smooth solutions ?
 A: These are the Cauchy-Riemann equations, but with respect to a different complex structure on the plane.
In more detail: the vector fields $A$ and $B$ uniquely determine a Riemannian metric on $\mathbb R^2$, by declaring $(A,B)$ to be an orthonormal frame. For any Riemannian metric in $2$d, in a neighborhood of each point there exist isothermal coordinates, that is, smooth coordinates $(x,y)$ in which the metric has the form $f(x,y)^2(dx^2 + dy^2)$ for some smooth positive function $f$. If necessary, we can interchange $x$ and $y$  so that these coordinates determine the same orientation as the frame $(A,B)$.
In these coordinates, both $(A,B)$ and $(f^{-1}\partial/\partial x, f^{-1}\partial/\partial y)$ are oriented orthornormal frames, so they are related (locally) by a rotation depending smoothly on the point: for some smooth function $\theta$,
\begin{align*}
A &= f(x,y)^{-1}\left(\cos\theta(x,y) \frac{\partial}{\partial x} - \sin\theta(x,y) \frac{\partial}{\partial y}\right),\\
B &= f(x,y)^{-1}\left(\sin\theta(x,y) \frac{\partial}{\partial x} + \cos\theta(x,y) \frac{\partial}{\partial y}\right).
\end{align*}
Then a little bit of linear algebra shows that the original pair of equations is equivalent to the Cauchy-Riemann equations in these coordinates.
