  We prove a general criterion to establish existence and uniqueness of a
short-time solution to an evolution equation involving "closed" sections of a
vector bundle, generalizing a method used recently by Bryant and Xu for
studying the Laplacian flow in G_2-geometry. We apply this theorem in balanced
geometry introducing a natural extension of the Calabi flow to the balanced
case. We show that this flow has always a unique short-time solution belonging
to the same Bott-Chern cohomology class of the initial balanced structure and
that it preserves the Kaehler condition. Finally we study explicitly the flow
on the Iwasawa manifold.
