Generalized Lyapunov theorem on Mal'tsev manifolds

The problem is studied of the extendability of a homomorphism $\mu\colon\Gamma\to G$, where $\Gamma$ is a lattice in a simply-connected nilpotent Lie group $N$, and $G$ is a linear algebraic group, to a homomorphism $\widetilde\mu\colon N\to G$ such that $\widetilde\mu|_\Gamma=\mu$. The case $\Gamma=\mathbf Z^n$ is considered in detail. The results obtained are applied to the study of reducibility of completely integrable equations on $N/\Gamma$.
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