Parameterizing and inverting analytic mappings with unit Jacobian

Let $x=(x_1,\dots,x_n)\in\mathbb{C}^n$ be a vector of complex variables, denote by $A=(a_{jk})$ a square matrix of size $n\geq 2$, and let $\varphi\in\mathcal{O}(\Omega)$ be an analytic function defined in a nonempty domain $\Omega\subset\mathbb{C}$. We investigate the family of mappings
$$ f=(f_1,\dots,f_n)\colon \mathbb{C}^n\rightarrow\mathbb{C}^n, \quad f[A,\varphi](x):=x+\varphi(Ax) $$
with the coordinates
$$ f_j \colon x \mapsto x_j + \varphi\biggl(\sum\limits_{k=1}^n a_{jk}x_k\biggr),\quad j=1,\dots,n, $$
whose Jacobian is identically equal to a nonzero constant for any $x$ such that all of $f_j$ are well defined.
Let $U$ be a square matrix such that the Jacobian of the mapping $f[U,\varphi](x)$ is a nonzero constant for any $x$ and moreover for any analytic function $\varphi\in\mathcal{O}(\Omega)$. We show that any such matrix $U$ is uniquely defined, up to a suitable permutation similarity of matrices, by a partition of the dimension $n$ into a sum of $m$ positive integers together with a permutation on $m$ elements.
For any $d=2,3,\dots$ we construct $n$-parametric family of square matrices $H(s), s\in\mathbb{C}^n$, such that for any matrix $U$ as above the mapping $x+\left((U\odot H(s))x\right)^d$ defined by the Hadamard product $U\odot H(s)$ has unit Jacobian. We prove any such mapping to be polynomially invertible and provide an explicit recursive formula for its inverse.