Nonlinear PDE for a 2D foliation 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.
 A: All foliations are locally diffeomorphic. So your equation depends on more than just a foliation. It also uses, most likely, the ambient Riemannian metric. You need to let us known what else it depends on (only the metric, perhaps only the affine connection of the metric, perhaps only the projective connection of the metric, perhaps only the conformal structure). Then we might recognize it.
A: 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.
