Yes, $\mathbf{x}=W(\lambda)\mathbf{1}$ is established form. But this equation forces numeric calculation to get $\lambda$ because of higher degree equation. On the other hand, main equation is also a non-linear simultaneous cubic equation. Theoretically, this is solvable. Just, this is my theme question.

Exactly, your statement is correct. On the contrary, the origin is that this consideration and constraint equations generate main equation eventually. Therefore, main equation must satisfy these conditions absolutely. Nevertheless, is it possible that we solve like $\mathbf{x}=\cdots$ ?