  We study spherically symmetric spacetimes in Einstein-aether theory in three
different coordinate systems, the isotropic, Painlev\`e-Gullstrand, and
Schwarzschild coordinates, in which the aether is always comoving, and present
both time-dependent and time-independent exact vacuum solutions. In particular,
in the isotropic coordinates we find a class of exact static solutions
characterized by a single parameter $c_{14}$ in closed forms, which satisfies
all the current observational constraints of the theory, and reduces to the
Schwarzschild vacuum black hole solution in the decoupling limit ($c_{14} =
0$). However, as long as $c_{14} \not= 0$, a marginally trapped throat with a
finite non-zero radius always exists, and in one side of it the spacetime is
asymptotically flat, while in the other side the spacetime becomes singular
within a finite proper distance from the throat, although the geometric area is
infinitely large at the singularity. Moreover, the singularity is a strong and
spacetime curvature singularity, at which both of the Ricci and Kretschmann
scalars become infinitely large.
