  In this paper, we study a coupled system of equations on oriented compact
4-manifolds which we call the Bach-Merkulov equations. These equations can be
thought of as the conformally invariant version of the classical
Einstein-Maxwell equations in general relativity. Inspired by the work of C.
LeBrun on Einstein-Maxwell equations on compact Kaehler surfaces, we give a
variational characterization of solutions to Bach-Merkulov equations as
critical points of the Weyl functional. We also show that extremal Kaehler
metrics are solutions to these equations, although, contrary to the
Einstein-Maxwell analogue, they are not necessarily minimizers of the Weyl
functional. We illustrate this phenomenon by studying the Calabi action on
Hirzebruch surfaces.
