  In this paper we study the existence of special symmetric solutions to a
Hamiltonian hyperbolic-hyperbolic coupled spin-field system, where the spins
are maps from $\mathbb R^{2+1}$ into the sphere $\S^2$ or the pseudo-sphere
$\H^2$. This model was introduced in \cite{Martina} and it is also known as the
{\it hyperbolic-hyperbolic generalized Ishimori system}. Relying on the
hyperbolic coordinates introduced in \cite{KNZ11}, we prove the existence of
equivariant standing waves in both regular hyperbolic coordinates as well as
similarity variables, and describe their asymptotic behaviour.
