Are there nonconstant solutions $U,V,W$ such that for all real $x,y$ we have
$$ U(x+y) = U(x) V(y) W(y) + U(y) V(x) W(y) + U(y) V(y) W(x) + U(y) V(x) W(x) + U(x) V(y) W(x) + U(x) V(x) W(y) $$
How to decide existance or uniqueness ?
Can we express the solutions with closed forms or differential equations ?