  We construct exact, regular and topologically non-trivial\ configurations of
the coupled Einstein-nonlinear sigma model in (3+1) dimensions. The ansatz for
the nonlinear $SU(2)$ field is regular everywhere and circumvents Derrick's
theorem because it depends explicitly on time, but in such a way that its
energy-momentum tensor is compatible with a stationary metric. Moreover, the
$SU(2)$ configuration cannot be continuously deformed to the trivial Pion
vacuum as it possesses a non-trivial winding number. We reduce the full coupled
4D Einstein nonlinear sigma model system to a single second order ordinary
differential equation. When the cosmological constant vanishes, such master
equation can be further reduced to an Abel equation. Two interesting regular
solutions correspond to a stationary traversable wormhole (whose only
\textquotedblleft exotic matter" is a negative cosmological constant) and a
(3+1)-dimensional cylinder whose (2+1)-dimensional section is a Lorentzian
squashed sphere. The Klein-Gordon equation in these two families of spacetimes
can be solved in terms of special functions. The angular equation gives rise to
the Jacobi polynomials while the radial equation belongs to the Poschl-Teller
family. The solvability of the Poschl-Teller problem implies non-trivial
quantization conditions on the parameters of the theory.
