On Some Relationships of Symmetric Sums: $u^n+v^n+w^n+(u+v+w)^n=k(u+v+w)(x^{n-1}+y^{n-1}+z^{n-1})$

Let $u, v, w, x, y, k$ and $z$ be any integers and suppose that $n$ is a given exponent. This study focuses on the interplay between sums of four powers and product of symmetric sums. In particular, the Diophantine equation $u^n+v^n+w^n+(u+v+w)^n=k(u+v+w)(x^{n-1}+y^{n-1}+z^{n-1})$ is introduced and partially characterized within the set of integers for exponent $n=3$. Moreover, this research formulates a conjecture for the equation presented in the title.