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Zero-curvature representations (ZCRs) are one of the main tools in the theoryof integrable $(1+1)$-dimensional PDEs. According to the preprintarXiv:1212.2199, for any given $(1+1)$-dimensional evolution PDE one can definea sequence of Lie algebras $F^p$, $p=0,1,2,3,\dots$, such that representationsof these algebras classify all ZCRs of the PDE up to local gauge equivalence.ZCRs depending on derivatives of arbitrary finite order are allowed.Furthermore, these algebras provide necessary conditions for existence ofBacklund transformations between two given PDEs. The algebras $F^p$ are definedin arXiv:1212.2199 in terms of generators and relations.  In the present paper, we describe some methods to study the structure of thealgebras $F^p$ for multicomponent $(1+1)$-dimensional evolution PDEs. Usingthese methods, we compute the explicit structure (up to non-essential nilpotentideals) of the Lie algebras $F^p$ for the Landau-Lifshitz, nonlinearSchrodinger equations, and for the $n$-component Landau-Lifshitz system ofGolubchik and Sokolov for any $n>3$. In particular, this means that for the$n$-component Landau-Lifshitz system we classify all ZCRs (depending onderivatives of arbitrary finite order), up to local gauge equivalence and up tokilling nilpotent ideals in the corresponding Lie algebras.  The presented methods to classify ZCRs can be applied also to other$(1+1)$-dimensional evolution PDEs. Furthermore, the obtained results can beused for proving non-existence of Backlund transformations between some PDEs,which will be described in forthcoming publications.

On Lie algebras responsible for zero-curvature representations of multicomponent (1+1)-dimensional evolution PDEs