I am trying to solve a 4th order nonlinear PDE for a real function $u(x,y)$ of two variables. It is too complicated to reproduce here but it exhibits the following two very nice properties:


1) if $u(x,y)$ is a solution, then $f(u(x,y))$ is also a solution **for any function** $f$.


2) if we think of the equation in the complex plane $z=x+iy$, then it is satisfied by **every (anti-)holomorphic function**, that is, $g(z)$ and $g(\bar{z})$ are solutions for any function $g$.


**Has anyone ever encountered such a nonlinear PDE?** If so, does it have a special name, and what are its known properties?


I am interested in finding as many real solutions as possible -- due to the nonlinearity, property 2) is not very useful in that regard. However, I've found a handful of solutions by inspection, and I suspect that the equation may in fact be integrable (though I am not sure how to test this), partly because of the above properties, and partly because of physical reasons.


*Note:* Property 1) has a nice geometric origin. Every function $u(x,y)$ defines a foliation of the plane by the level sets $u(x,y)=$ constant. We can then think of $u$ as a coordinate labeling the "leaves" in the foliation, and the change $u\to f(u)$ is just a relabeling of coordinates (as long as $f$ is monotonic).


In other words, this PDE can be thought of as an equation for a foliation. Any given foliation can be represented by many functions $u(x,y)$, all of which are functions of each other, and property 1) is just the statement of reparameterization invariance.



Nonlinear PDE for a 2D foliation
It's easy to derive a *third*-order (nonlinear) differential equation for $u(x,y)$ that satisfies your conditions (1) and (2): Namely, set $\theta(x,y) = \arctan\bigl(u\_y(x,y)/u\_x(x,y)\bigr)$ and then consider
$$
\theta\_{xx} + \theta\_{yy} = 0.
$$
When one expresss this equation explicitly in terms of the partial derivatives of $u$, one obtains a third-order nonlinear polynomial PDE whose solutions have exactly the properties (1) and (2). (However, I'm actually interpreting (2) to mean that the *real part* of $g$ satisfies your equation, since you started out stating that the equation was for a *real* function $u(x,y)$, while a nonconstant holomorphic function $g(z)$ is never real-valued. Maybe you have a typo somewhere? Clarification of this point would help.) 


The general solution of this third order equation depends on three arbitrary functions of one variable.


Now, your fourth-order equation *could* be a (combination of) derivative(s) of this third-order equation. If it is suitably non-degenerate its general solution will depend on four functions of one variable, so it will have solutions that don't satisfy this third-order equation, in which case, there will be solutions that aren't of the special form you have found so far.


