  We present an equivalent criterion for the global existence of Euler's
multiplier for an integrable one-form taking into account the corresponding
codim-1-foliation. In particular, the impact of inseparable leaves is
considered. Here, we suppose that the foliation can be reduced to a graph; we
also discuss obstructions for this assumption. The properties of this graph are
decisive for the global existence of Euler's multiplier. As applications we
investigate some special cases in which the graph turns out to look very
simple.
