Obvious Killing vectors? What obvious Killing vectors do these metrics have?
(a) $ds^2=\frac{1}{y^2}(dx^2+dy^2)$,  $-\infty<x<\infty,y>0$
(b) $ds^2=d\mathscr{X}^2+\sinh^2{\mathscr{X}}d\phi^2$,  $0\leq\mathscr{X}<\infty,0\leq\phi<2\pi$
If someone could help me out identifying the $obvious$ Killing vectors, and explain how to do so, I would be eternally grateful.

The horizontal translation $(x,y)\mapsto (x+c,y)$ preserves (a), so $\frac{\partial}{\partial x}$ is Killing vector field of (a).
Similarly, $(\mathscr{X},ϕ)↦(\mathscr{X},ϕ+c)$ preserves (b), so...
