= Solution
An <asymptotically flat spacetime> is a <stationary spacetime> if it admits a complete stationary <Killing vector field> $K$, with $\mathcal L_Kg=0$, timelike near infinity and normalized there to a unit time translation. It need not remain timelike in an <ergoregion of a stationary spacetime>. It is axisymmetric if it admits an axial <Killing vector field> $\Phi$, whose flow is a rotational $U(1)$ action: its spacelike orbits away from the axis are closed with period $2\pi$, and it vanishes on the regular rotation axis. For a <stationary axisymmetric spacetime>, the generators can be chosen to commute, $[K,\Phi]=0$. These are geometric symmetries, not merely time-independent or angle-independent coordinate labels.
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