 Solution

The body is invariant under the reflections and half-turns that preserve its $z$-axis. A polar vector force $F{\mathbf{e}}_z$ can therefore produce only the polar velocity $U_z{\mathbf{e}}_z$; every rotation is excluded because angular velocity is a pseudovector. Hence only $U_z$ is nonzero when $\mathbf{F}=(0,0,F)$ and $\mathbf{G}=0$.

For $\mathbf{F}=(F,0,0)$, the surviving symmetry-allowed components arewhile $U_y,U_z,{\Omega }_x,{\Omega }_z$ vanish. The coupling between translation in $x$ and rotation about $y$ is permitted because the two horizontal rods lie at opposite vertical offsets.