Abstract
The complete list of the third-order polynomial evolution equations of not normal type with nontrivial Lie-Bäcklund symmetries is obtained. Almost all the equations appearing in the list have the hereditary recursion operator of Lie-Bäcklund symmetries so that the Lie algebras of their Lie-Bäcklund symmetries are infinite dimensional and commutative. Further, two fifth-order equations of not normal type admitting seventh-order Lie-Bäcklund symmetry are presented. They do not belong to the hierarchies of the third-order equations derived in the classification.
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