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For any (1+1)-dimensional (multicomponent) evolution PDE, we define asequence of Lie algebras $F^p$, $p=0,1,2,3,...$, which are responsible for allLax pairs and zero-curvature representations (ZCRs) of this PDE.  In our construction, jets of arbitrary order are allowed. In the case oflower order jets, the algebras $F^p$ generalize Wahlquist-Estabrookprolongation algebras.  To achieve this, we find a normal form for (nonlinear) ZCRs with respect tothe action of the group of gauge transformations. One shows that any ZCR islocally gauge equivalent to the ZCR arising from a vector field representationof the algebra $F^p$, where $p$ is the order of jets involved in the $x$-partof the ZCR.  More precisely, we define a Lie algebra $F^p$ for each nonnegative integer$p$ and each point $a$ of the infinite prolongation $E$ of the evolution PDE.So the full notation for the algebra is $F^p(E,a)$.  Using these algebras, one obtains a necessary condition for two givenevolution PDEs to be connected by a Backlund transformation.  In this paper, the algebras $F^p(E,a)$ are computed for some PDEs of KdVtype. In a different paper with G. Manno, we compute $F^p(E,a)$ formulticomponent Landau-Lifshitz systems of Golubchik and Sokolov. Among theobtained Lie algebras, one encounters infinite-dimensional algebras of certainmatrix-valued functions on some algebraic curves. Besides, some solvable idealsand semisimple Lie algebras appear in the description of $F^p(E,a)$.  Applications to classification of KdV and Krichever-Novikov type equationswith respect to Backlund transformations are also briefly discussed.

Higher jet prolongation Lie algebras and Backlund transformations for (1+1)-dimensional PDEs