Polynomial conservation laws and exact solutions associated with isometric and homothetic symmetries in the nonlinear sigma model

In the nonlinear $\sigma$ model, conserved tensor currents associated with the presence of isometric, homothetic, and affine motions in the space of values of the chiral field $V^N$ are constructed. New classes of exact solutions in the $SO(3)$- and $SO(5)$-invariant $\sigma$ models are obtained using the connection between the groups of isometric and homothetic motions of space-time and the isometric motions in $V^N$. Some methods for obtaining exact solutions in the four-dimensional $\sigma$ model with nontrivial topological charge are considered.