  We consider a class of ans\"atze for the construction of exact solutions of
the Einstein-nonlinear $\sigma$-model system with an arbitrary cosmological
constant in (3+1) dimensions. Exploiting a geometric interplay between the
$SU(2)$ field and Killing vectors of the spacetime reduces the matter field
equations to a single scalar equation (identically satisfied in some cases) and
simultaneously simplifies Einstein's equations. This is then exemplified over
various classes of spacetimes, which allows us to construct stationary black
holes with a NUT parameter and uniform black strings, as well as time-dependent
solutions such as Robinson-Trautman and Kundt spacetimes, Vaidya-type radiating
black holes and certain Bianchi~IX cosmologies. In addition to new solutions,
some previously known ones are rederived in a more systematic way.
