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The integrability of the\ $\Lambda-$Einstein-nonlinear $SU(2)$ $\sigma$-modelwith nonvanishing cosmological charge is studied. We apply the method ofsingularity analysis of differential equations and we show that the equationsfor the gravitational field are integrable. The first few terms of the solutionare presented.

Integrability of the Einstein-nonlinear $SU(2)$ $\sigma$-model in a nontrivial topological sector