  In the present work we analyze all the possible spherically symmetric
exterior vacuum solutions allowed by the Einstein-Aether theory with static
aether. We show that there are four classes of solutions corresponding to
different values of a combination of the free parameters, $c_{14}=c_1+c_4$,
which are: $ 0 < c_{14}<2$, $c_{14} < 0$, $c_{14}=2$ and $c_{14}=0$. We present
explicit analytical solutions for $c_{14}=3/2, 16/9, 48/25, -16, 2$ and $0$.
The first case has some pathological behavior, while the rest have all
singularities at $r=0$ and are asymptotically flat spacetimes. For the
solutions $c_{14}=16/9, 48/25\, \mathrm{\, and \,}\, -16$ we show that there
exist no horizons, neither Killing nor universal horizon, thus we have naked
singularities. Finally, the solution for $c_{14}=2$ has a metric component as
an arbitrary function of radial coordinate, when it is chosen to be the same as
in the Schwarzschild case, we have a physical singularity at finite radius,
besides the one at $r=0$. This characteristic is completely different from
General Relativity.
