roots of system of nonlinear equations

Question

I can't get any solutions beside when $x=0\vee y=0 \vee z=0$ $$yz-2x\lambda-2x\mu=0\tag{1}$$ $$xz-2y\mu=0\tag{2}$$ $$xy-4z\lambda =0\tag{3}$$ $$x^2+y^2=4\tag{4}$$ $$x^2+2z^2=3\tag{5}$$ Can you help me? I seem to be doing algebraic manipulation that lead me nowhere.

Answer 1

From (4) and (5) we obtain:

$$y^2z^2=(4-x^2)(3-x^2)/2......(6)$$

From (1) we obtain: $$y^2z^2=4x^2(\lambda+\mu)^2......(7)$$

From (6)-(7) we have: $$(4-x^2)(3-x^2)=8x^2(\lambda+\mu)^2...(8)$$

We can then solve $x^2$ from (8).

Once $x$ is solved, we can then solve other equations (4) and (5) for $y$ and $z$...