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

the journal of high energy physics/the journal of high energy physics

Extensions of the generalized hedgehog ansatz for the Einstein-nonlinear  $\sigma$-model system: black holes with NUT, black strings and time-dependent  solutions