= Solution
Integrating the two <vector fields> gives their complete <flow maps>:
$$
\boxed{\phi_t(x,y,z)=(x\cos t-y\sin t,\ x\sin t+y\cos t,\ e^t z),\qquad \psi_t(x,y,z)=(e^t x,e^t y,e^t z).}
$$
The first is a simultaneous <rotation about the z-axis> through angle $t$ and a dilation by $e^t$ in the $z$-direction. The second is an isotropic dilation about the origin. Both satisfy the group law for all real $t$ and have inverse given by time $-t$. In particular $\phi_t$ is not a pure rotation unless restricted to the plane $z=0$.
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