  How do the global properties of a Lorentzian manifold change when endowed
with a vector field? This interesting question is tackled in this paper within
the framework of Einstein-Aether (EA) theory which has the most general
diffeomorphism-invariant action involving a spacetime metric and a vector
field. After classifying all the possible nine vacuum solutions with and
without cosmological constant in Friedmann-Lema{\^{\i}}tre-Robertson-Walker
(FLRW) cosmology, we show that there exist three singular solutions in the EA
theory which are not singular in the General Relativity (GR), all of them for
$k=-1$, and another singular solution for $k=1$ in EA theory which does not
exist in GR. This result is cross-verified by showing the focusing of timelike
geodesics using the Raychaudhuri equation. These new singular solutions show
that GR and EA theories can be completely different, even for the FLRW
solutions when we go beyond flat geometry ($k=0$). In fact, they have different
global structures. In the case where $\Lambda=0$ ($k=\pm 1$) the vector field
defining the preferred direction is the unique source of the curvature.
