Solution of functional equations related to elliptic functions. III

Let $s,m\in {\Bbb N}$, $s\ge 2$. We solve the functional equation
$$ f_1(x_1+z)\ldots f_{s-1}(x_{s-1}+z)f_s(x_1+\ldots +x_{s-1}-z) = \sum_{j=1}^{m} \varphi_j(x_1,\ldots,x_{s-1})\psi_j(z), $$
for unknown entire functions $f_1,\ldots,f_s:{\Bbb C}\to {\Bbb C}$, $\varphi_j: {\Bbb C}^{s-1}\to {\Bbb C}$, $\psi_j: {\Bbb C}\to {\Bbb C}$ in the case of $s\ge 3$, $m\le 2s-1$. All non-elementary solutions are described by the Weierstrass sigma-function. Previously, such results were known for $m\le s+1$. The considered equation arises in the study of polylinear functional-differential operators and multidimensional vector addition theorems.