Find, using a Lagrange multiplier, the four stationary points in $\mathbb{R}^{3}$ of the function $x^{2}+y^{2}+z^{2}$ subject to the constraint $x^{2}+2 y^{2}-z^{2}=1$. By considering the situation geometrically, or otherwise, identify the nature of the constrained stationary points.

How would your answers differ if, instead, the stationary points of the function $x^{2}+2 y^{2}-z^{2}$ were calculated subject to the constraint $x^{2}+y^{2}+z^{2}=1 ?$