Solution of functional equations related to elliptic functions. II

Let $s,m, d\in \mathbb{N}$, $s\ge 2$. We solve the functional equation
\begin{gather*} f_1(\mathbf{u}_1+\mathbf{v})\ldots f_{s-1}(\mathbf{u}_{s-1}+\mathbf{v})f_s(\mathbf{u}_1+\ldots +\mathbf{u}_{s-1}-\mathbf{v}) \\ =\sum_{j=1}^{m} \phi_j(\mathbf{u}_1,\ldots,\mathbf{u}_{s-1})\psi_j(\mathbf{v}), \end{gather*}
for unknown entire functions $f_1,\ldots,f_s:\mathbb{C}^d\to \mathbb{C}$, $\phi_j: (\mathbb{C}^d)^{s-1}\to \mathbb{C}$, $\psi_j: \mathbb{C}^d\to \mathbb{C}$ in the case of $m\le s+1$. All non-elementary solutions are described by the Weierstrass sigma-function. Previously, such results were known only for $s=2$, $m=1,2$, as well as for $d=1$, $s=2,3$. The considered equation arises in the study of polylinear functional-differential operators and multidimensional vector addition theorems.