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.
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