# Compatible solution of PDE

Let $$c=c(z, \bar z)$$ be a complex function satisfying $$\partial_{z} \bar c=\partial_{\bar z} c$$. It follows that there exists a real function $$f$$ such that $$\partial_{\bar z} f=-c$$. Would it be possible to find a real function $$g$$ such that $$\partial_{\bar z} g=-i c$$ is satisfied where $$i^2=-1$$?

If such a $$g$$ exists, then $$f+ig$$ is holomorphic, so $$f$$ is harmonic, so $$\partial_z c=0$$, i.e. $$c$$ is locally constant. But if $$c$$ is locally constant, then $$f$$ is harmonic, and so locally the real part of a holomorphic function, with imaginary part $$g$$, so $$g$$ exists locally.