  Zero-curvature representations (ZCRs) are one of the main tools in the theory
of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be
interpreted as ZCRs.
  For any (1+1)-dimensional scalar evolution equation $E$, we define a family
of Lie algebras $F(E)$ which are responsible for all ZCRs of $E$ in the
following sense. Representations of the algebras $F(E)$ classify all ZCRs of
the equation $E$ up to local gauge transformations. To achieve this, we find a
normal form for ZCRs with respect to the action of the group of local gauge
transformations.
  As we show in other publications, using these algebras, one obtains some
necessary conditions for integrability of the considered PDEs (where
integrability is understood in the sense of soliton theory) and necessary
conditions for existence of a B\"acklund transformation between two given
equations. Examples of proving non-integrability and applications to obtaining
non-existence results for B\"acklund transformations are presented in other
publications as well.
  In our approach, ZCRs may depend on partial derivatives of arbitrary order,
which may be higher than the order of the equation $E$. The algebras $F(E)$
generalize Wahlquist-Estabrook prolongation algebras, which are responsible for
a much smaller class of ZCRs.
  In this paper we describe general properties of $F(E)$ and present generators
and relations for these algebras. In other publications we study the structure
of $F(E)$ for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt,
Sawada-Kotera types. Among the obtained algebras, one finds
infinite-dimensional Lie algebras of certain matrix-valued functions on
rational and elliptic algebraic curves.
