What is an example of a Lie group $G$ not ismorphic to the Poincare upper half space $H^n$  but satisfy the following:

For every left invariant metric $g$ we draw all geodesics. Then we CAN rescale $g$ conformally to a metric $h$ and we observe that at each point $x\in G$
all drown half ray $g$- geodesics initiating $x$ have finite $h$-length except a geodesic tangent to a unique direction