  We continue our study of symplectically flat bundles. We broaden the notion
of symplectically flat connections on symplectic manifolds to $\zeta$-flat
connections on smooth manifolds. These connections on principal bundles can be
represented by maps from an extension of the base's fundamental group to the
structure group. We introduce functionals with zeroes being symplectically flat
connections and study their critical points. Such functionals lead to novel
geometric flows. We also describe some characteristic classes of the
$\zeta$-flat bundles.
