To understand some physics problem, I want to know if there is (**non-$L^2$** or $L^2$) solution to the Seiberg-Witten equation on $\mathbb{R}^4$
\begin{equation}
{D}_A \psi = 0\\
F_A^+ = i\zeta\omega + \sigma(\psi)
\end{equation}
where $\zeta$ is a real number, and $\omega$ is the standard symplectic form on $\mathbb{R}^4$.

I know that when $\zeta = 0$, a square-integrable solution would imply $\psi = 0$ everywhere, because by Weitzenbock formula \begin{equation} D_A^*{D_A}\psi = \nabla _A^*{\nabla _A}\psi + \frac{s}{4}\psi + \frac{1}{2}F_A^ + \psi \end{equation} and then using the S-W equation itself, one gets \begin{equation} \int {{{\left| {{\nabla _A}\psi } \right|}^2} + \frac{1}{4}{{\left| \psi \right|}^4} + \left( {i\zeta\omega \cdot \psi ,\psi } \right)} = - \frac{1}{4}\int {s{{\left| \psi \right|}^2}} \end{equation} which forces $\psi = 0$ when $s = \zeta = 0$

But I wonder what happens if $\zeta \ne 0$?

Is there known solution (non-$L^2$ is also good) to the equation? Any reference would be great.