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.