  We use static solutions of 5-dimensional Kaluza-Klein gravity to generate
several classes of static, spherically symmetric spacetimes which are analytic
solutions to the equation $^{(4)}R = 0$, where $^{(4)}R$ is the
four-dimensional Ricci scalar. In the Randall $&$ Sundrum scenario they can be
interpreted as vacuum solutions on the brane. The solutions contain the
Schwarzschild black hole, and generate new families of traversable Lorenzian
wormholes as well as nakedly singular spacetimes. They generalize a number of
previously known solutions in the literature, e.g., the temporal and spatial
Schwarzschild solutions of braneworld theory as well as the class of self-dual
Lorenzian wormholes. A major departure of our solutions from Lorenzian
wormholes {\it a la} Morris and Thorne is that, for certain values of the
parameters of the solutions, they contain three spherical surfaces (instead of
one) which are extremal and have finite area. Two of them have the same size,
meet the "flare-out" requirements, and show the typical violation of the energy
conditions that characterizes a wormhole throat. The other extremal sphere is
"flaring-in" in the sense that its sectional area is a local maximum and the
weak, null and dominant energy conditions are satisfied in its neighborhood.
After bouncing back at this second surface a traveler crosses into another
space which is the double of the one she/he started in. Another interesting
feature is that the size of the throat can be less than the Schwarzschild
radius $2 M$, which no longer defines the horizon, i.e., to a distant observer
a particle or light falling down crosses the Schwarzschild radius in a finite
time.
