Let \(K\) be a connected compact Lie group, \(G\) its complexification, \(X\) a compact Kähler manifold, and \(E\to X\) be a principal holomorphic \(G\)-bundle over \(X\). Let \(W\) be a complex vector space and \(\rho: K\to U(W)\) a unitary representation of \(K\), which lifts to a representation \(\widetilde\rho\) of \(G\) and let \(V\to X\) be the associated vector bundle to \(\widetilde\rho: V=E \times_{\tilde\rho}W\). The author investigates a class of equations for a connection \(A\) on the principal bundle \(E\) and a section \(\Phi\) of \(V\) over \(X\). This class, as the author shows, contains, as special cases, Hermitian-Einstein equation, vortex equation, Vafa-Witten equations as well as non-abelian monopole equations. For the solvability of the equations the author introduces the notion of \(\tau\)-stability condition and that of indecomposability for the pair \((E,\Phi)\), and proves that for an indecomposable pair \((E,\Phi)\), the equation is satisfied if and only if \((E,\Phi)\) is \(\tau\)-stable. The proof is considerably hard to follow, since it quotes a number of results due to S. B. Bradlow, S. K. Donaldson and others.