Let $f$ be any complex function on $\mathbb{CP}^1$. Denote the local coordinates of $\mathbb{CP}^1$ as $z,\bar{z}$. Does the following equation

$\frac{\partial}{\partial z} f = \frac{\partial}{\partial \bar{z}} f = \frac{i}{2r} f$

where $r$ is the radius of $\mathbb{CP}^1$, have solution? If yes, then is there an easy example?

Let $f$ be any complex function on $\mathbb{CP}^1$. Denote the local coordinates of $\mathbb{CP}^1$ as $z,\bar{z}$. Does the following equation

$\frac{\partial}{\partial z} f = \frac{\partial}{\partial \bar{z}} f = \frac{i}{2r} f$

where $r$ is the radius of $\mathbb{CP}^1$, have nontrivial solutions? If yes, then is there an easy example?