We begin by recalling the definition of teichmuller space but stated a little more convolutedly (which will make it easy to generalize).

Given a surface $S$ we can define teichmuller space $T(S)$ to be the set of all complex structures on $S$ modulo homeomorphisms isotopic to the identity (that is an atlas $A$ of charts $\mathbb{R}^2 \rightarrow S $ such that if $u_i, u_j$ are two charts that overlap on some domain $D$ then $u_i \circ u_j^{-1}$ is a holomorphic function $D \rightarrow E$ where $D,E \subseteq \mathbb{R}^2$

Now a holomorphic function $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ is defined as a function $u(x,y), v(x,y)$ which obeys the Cauchy Riemann System of PDEs

$$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$$

So now naturally the question then arises, suppose we consider some OTHER system of PDEs and leave the rest of the definition of teichmuller space unchanged, ex we swap out the cauchy riemann equations with some other system which could be as arbitrary as the following:

$$ xy\frac{\partial u}{\partial x} = e^u \frac{\partial v^2}{\partial y^2} \\ \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} = 1$$

Then in that case, how does the resultant "generalized-teichmuller" space of our surface change?

I'm interested in results for any non cauchy-riemann system of equations (perhaps even just antiholomorphic or harmonic would be a good place to begin or just generally conformal)

We begin by recalling the definition of Teichmüller space but stated a little more convolutedly (which will make it easy to generalize).

Given a surface $S$ we can define Teichmüller space $T(S)$ to be the set of all complex structures on $S$ modulo homeomorphisms isotopic to the identity (that is an atlas $A$ of charts $\mathbb{R}^2 \rightarrow S $ such that if $u_i$, $u_j$ are two charts that overlap on some domain $D$ then $u_i \circ u_j^{-1}$ is a holomorphic function $D \rightarrow E$ where $D,E \subseteq \mathbb{R}^2$.

Now a holomorphic function $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ is defined as a function $(u(x,y), v(x,y))$ which obeys the Cauchy–Riemann System of PDEs

$$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}.$$

So now naturally the question then arises, suppose we consider some OTHER system of PDEs and leave the rest of the definition of Teichmüller space unchanged, e.g., we swap out the Cauchy–Riemann equations with some other system which could be as arbitrary as the following:

$$ xy\frac{\partial u}{\partial x} = e^u \frac{\partial v^2}{\partial y^2} \\ \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} = 1.$$

Then in that case, how does the resultant "generalized-Teichmüller" space of our surface change?

I'm interested in results for any non-Cauchy–Riemann system of equations (perhaps even just antiholomorphic or harmonic would be a good place to begin or just generally conformal).