ca.analysis and odes - Transforming a multivariable integral to make it separable

In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine that such requirements are stated correctly.

Let's say that a function $F : mathbb{C}^ntomathbb{C}$ separates (where $mathbb{C}$ is the complex numbers) if we can factorize it like
$F(z_1,ldots,z_n)=F_1(z_1)timescdotstimes F_n(z_n)$. In this case we can factorize the integral of $F$ like $iiint F(z_1,ldots,z_n) ~dz_1cdots dz_n = int F_1(z_1)dz_1times cdots timesint F_n(z_n)dz_n$.

In case $F$ does not separate, we might be able to change variables to make it separate. If $g : mathbb{C}^ntomathbb{C}^n$ is nice enough and $Delta$ is its Jacobian determinant (or maybe its inverse depending on which way you like to define it), then
$iiint F(z_1,ldots,z_n) ~dz_1cdots dz_n = iiint G(w_1,ldots,w_n) ~dw_1cdots dw_n$, where $G(w_1,ldots,w_n)=F(g(w_1,ldots,w_n))Delta(w_1,ldots,w_n)^{-1}$.

My question is: for which $F$ can $g$ be chosen so that $G$ separates?

An example everyone knows is $F(z_1,ldots,z_n)=exp(Q(z_1,ldots,z_n))$, where $Q$ is a quadratic form. Then $g$ can be chosen to be a linear transformation that diagonalizes the quadratic form. In general, a non-linear transformation will be required.