Let C be the curve at the intersection of two level surfaces M(x, y, z) = 5 and N(x,y,z) = 0, passing through a point P (1,1,1). Let $M(x, y, z) = 2x^2 + y^2 + 2z^2$ and $N(x, y, z) = xy-z$. Find curve C in the parametric form <x(t), y(t), z(t)>. I am quite stuck on this one. I could let $z = xy$...

Consider the (four) one-dimensional representations of $V_4\simeq S_2\times S_2$ the Klein four group, as listed for example here. It is convenient to consider these as four vectors forming a basis for $\mathbb{Q}^4$ (no need for complex here): $$\left\{\begin{bmatrix}1\\1\\1\\1\end{bmatrix},\b...