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

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. In [arXiv:1303.3575], for any (1+1)-dimensional scalar evolution equation $E$, we defined 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. In [arXiv:1804.04652] we showed that, using these algebras, one obtains necessary conditions for existence of a B\"acklund transformation between two given equations. The algebras $F(E)$ are defined in terms of generators 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 in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs of order $5$. Also, we prove a result announced in [arXiv:1303.3575] on the structure of the algebras $F(E)$ for certain classes of equations of orders $3$, $5$, $7$, which include KdV, mKdV, Kaup-Kupershmidt, Sawada-Kotera type equations. Among the obtained algebras for equations considered in this paper and in [arXiv:1804.04652], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curves of genus $1$ and $0$. In this 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.