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Zero-curvature representations (ZCRs) are one of the main tools in the theoryof integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can beinterpreted as ZCRs. In [arXiv:1303.3575], for any (1+1)-dimensional scalarevolution equation $E$, we defined a family of Lie algebras $F(E)$ which areresponsible for all ZCRs of $E$ in the following sense. Representations of thealgebras $F(E)$ classify all ZCRs of the equation $E$ up to local gaugetransformations. In [arXiv:1804.04652] we showed that, using these algebras,one obtains necessary conditions for existence of a B\"acklund transformationbetween two given equations. The algebras $F(E)$ are defined in terms ofgenerators and relations. In this paper we show that, using the algebras$F(E)$, one obtains some necessary conditions for integrability of(1+1)-dimensional scalar evolution PDEs, where integrability is understood inthe sense of soliton theory. Using these conditions, we prove non-integrabilityfor some scalar evolution PDEs of order $5$. Also, we prove a result announcedin [arXiv:1303.3575] on the structure of the algebras $F(E)$ for certainclasses of equations of orders $3$, $5$, $7$, which include KdV, mKdV,Kaup-Kupershmidt, Sawada-Kotera type equations. Among the obtained algebras forequations considered in this paper and in [arXiv:1804.04652], one findsinfinite-dimensional Lie algebras of certain polynomial matrix-valued functionson affine algebraic curves of genus $1$ and $0$. In this approach, ZCRs maydepend on partial derivatives of arbitrary order, which may be higher than theorder of the equation $E$. The algebras $F(E)$ generalize Wahlquist-Estabrookprolongation algebras, which are responsible for a much smaller class of ZCRs.

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs