Symmetries, conservation laws, and exact solutions in nonlinear sigma models

Nonlinear four-dimensional $\sigma$ models interacting with the gravitational field are considered. On the basis of the connection between the groups of isometric motions of the space-time $V_4$ and the space $V_N$ of values of the chiral field exact solutions are obtained for the Einstein and chiral-field equations for spherical and plane symmetry. The conserved currents associated with the presence of the isometric motions in $V_N$ are constructed.