  In this paper we show that the dimensionally reduced Seiberg-Witten equations
lead to a Higgs field and study the resulting moduli spaces. The moduli space
arising out of a subset of the equations, shown to be non-empty for a compact
Riemann surface of genus g >= 1, gives rise to a family of moduli spaces
carrying a hyperkahler structure. For the full set of equations the
corresponding moduli space does not have the aforementioned hyperkahler
structure but has a natural symplectic structure. For the case of the torus,
g=1, we show that the full set of equations has a solution, different from the
``vortex solutions''.
