  Let $X=\mathbb{C}\times\Sigma$ be the product of the complex plane and a
compact Riemann surface. We establish a classification theorem of solutions to
the Seiberg-Witten equation on $X$ with finite analytic energy. The spin bundle
$S^+\to X$ splits as $L^+\oplus L^-$. When $2-2g\leq c_1(S^+)[\Sigma]<0$, the
moduli space is in bijection with the moduli space of pairs
$((L^+,\bar{\partial}), f)$ where $(L^+,\bar{\partial})$ is a holomorphic
structure on $L^+$ and $f: \mathbb{C}\to H^0(\Sigma, L^+,\bar{\partial})$ is a
polynomial map. Moreover, the solution has analytic energy $-4\pi^2d\cdot
c_1(S^+)[\Sigma]$ if $f$ has degree $d$.
  When $c_1(S^+)=0$, all solutions are reducible and the moduli space is the
space of flat connections on $\bigwedge^2 S^+$.
  We also estimate the decay rate of these solutions at infinity.
