Inverse Jacobi multiplier

Given a vector field $Z\in \mathfrak{X}(M)$ on a $n$-dimensional manifold $M$ with a distinguished volume form $\mathrm{\Omega }$, an inverse Jacobi multiplier (or inverse multiplier) is a smooth function $\mathrm{\Delta }$ such that

or

See @berrone2003inverse page 14. But also, this is the definition given in @hu2015inverse for inverse integrating factor.

It is satisfied that if $\mathrm{\Delta }$ is an inverse Jacobi multiplier then $1/\mathrm{\Delta }$ is a Jacobi last multiplier.

It can be shown that the product of symmetrising factors is an inverse Jacobi multiplier. See the paper "symfactor", or the video 003 and xournal 199.