Groupclassificationofquasilinearthird-orderevolutionequationsisgivenbyusingtheclassicalinfinitesimalLiemethod,thetechniqueofequivalencetransfor-mations,andthetheoryofclassificationofabstractlow-dimensionalLiealgebras.WeshowthattherearethreeequationsadmittingsimpleLiealgebrasofdimensionthree.Allnon-equivalentequationsadmittingsimpleLiealgebrasarenothingbutthesethree.Furthermore,wealsoshowthatthereexisttwo,five,twenty-nineandtwenty-sixnon-equivalentthird-ordernonlinearevolutionequationsadmittingone-,two-,three-,andfour-dimensionalsolvableLiealgebras,respectively.
