Singular Liouville fields and spiky strings in ${\text{rr}}^{1,2}$ and $SL(2,\text{rr})$

The closed string dynamics in ${\text{rr}}^{1,2}$ and $SL(2,\text{rr})$ is studied within the scheme of Pohlmeyer reduction. In both spaces two different classes of string surfaces are specified by the structure of the fundamental quadratic forms. The first class in ${\text{rr}}^{1,2}$ is associated with the standard lightcone gauge strings and the second class describes spiky strings and their conformal deformations on the Virasoro coadjoint orbits. These orbits correspond to singular Liouville fields with the monodromy matrixes $\pm I$. The first class in $SL(2,\text{rr})$ is parameterized by the Liouville fields with vanishing chiral energy functional. Similarly to ${\text{rr}}^{1,2}$, the second class in $SL(2,\text{rr})$ describes spiky strings, related to the vacuum configurations of the $SL(2,\text{rr})\mathrm{/}U(1)$ coset model.