# Solutions to nonlinear PDE derived from the Dirichlet energy in the stereographic plane

Define a complex valued function $$z$$ in the stereographic plane and let $$z=z(s,s^{*})$$ for $$s$$ a complex valued variable and $$*$$ denoting the complex conjugate.

Define the Dirichlet energy as

$$\int \frac{\lvert z_s \rvert^{2} + \lvert z_{s^{*}} \rvert^{2}}{(1 + \lvert z \rvert^{2})^{2}} \ ds ds^*.$$

Minimizing this with respect to $$z^*$$ we find that $$z$$ must satisfy the following nonlinear PDE:

$$(1 + \lvert z \rvert^{2}) z_{s s^*} = 2 z^{*} z_s z_{s^{*}}.$$

Some simple solutions to this are $$z = f(s)$$ or $$z = g(s^{*})$$ for arbitrary functions $$f, g$$.

Are there other known solutions to this equation?

Edit added: Are these solutions related to the spherical harmonics? If so, how can we compose two solutions in the stereographic plane to find another solution?

I see reference to this equation here, but cannot find other useful references. Any tips on work that has been done on this equation are appreciated.