Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a power series expansion in $z,\bar z$ at every point $z \neq z_0$ in $\mathbb C/(\mathbb Z+i \mathbb Z).$
Now, consider the operator $\partial_{\bar z}= \partial_x + i \partial_y$ where $z= x+iy$. I ask:
Does there always exist for any $\lambda \in \mathbb C$ a non-zero solution to the PDE $$(\partial_{\bar z} + \lambda)f(z)=0 \text{ for } z \in \mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\}$$ in the set of functions above?
If I would not allow to take out the point $z_0$, then the answer would be "No". In this case, there would only be a discrete set of points $\lambda$ for which the PDE has a solution. This follows easily from some Fourier representation of the function $f$. But since I now allow for some singular behaviour around $z_0$, it is no longer clear to me.
