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 ofintegrable PDEs. In [13], for any (1+1)-dimensional scalar evolution equationE, wedefined a family of Lie algebrasF(E) which are responsible for all ZCRs ofEin thefollowing sense. Representations of the algebrasF(E) classify all ZCRs of the equationEup to local gauge transformations. In [12] we showed that, using these algebras, oneobtains necessary conditions for existence of a Bäcklund transformation between twogiven equations.In this paper we show that, using the algebrasF(E), one obtains some necessaryconditions for integrability of (1+1)-dimensional scalar evolution PDEs, where integra-bility is understood in the sense of soliton theory. Using these conditions, we provenon-integrability for some scalar evolution PDEs of order 5. Also, we prove a resultannounced in [13] on the structure of the algebrasF(E) for certain classes of equationsof orders 3, 5, 7, which include KdV, mKdV, Kaup–Kupershmidt, Sawada–Kotera typeequations. Among the obtained algebras for equations considered in this paper andin [12], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valuedfunctions on affine algebraic curves of genus 1 and 0