  We study generalisations to the structure groups U(n) of the familiar
(abelian) Seiberg-Witten monopole equations on a four-manifold $X$ and their
moduli spaces. For $n=1$ one obtains the classical monopole equations. For $n >
1$ our results indicate that there should not be any non-trivial
gauge-theoretical invariants which are obtained by the scheme `evaluation of
cohomology classes on the fundamental cycle of the moduli space'. For, if
$b_2^+$ is positive the moduli space should be `cobordant' to the empty space
because we can deform the equations so as the moduli space of the deformed
equations is generically empty. Furthermore, on K\"ahler surfaces with $b_2^+ >
1$, the moduli spaces become empty as soon as we perturb with a non-vanishing
holomorphic 2-form.
