@misc{TsuchidaWolf, author = {Tsuchida, Takayuki and Wolf, Thomas}, title = {Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8391}, number = {05-05}, abstract = {We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov--Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of \$2^{\mbox{\scriptsize nd }}\$order systems with a \$3^{\mbox{\scriptsize rd }}\$order or a \$4^{\mbox{\scriptsize th }}\$order symmetry and \$3^{\mbox{\scriptsize rd }}\$order systems with a \$5^{\mbox{\scriptsize th }}\$order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.}, language = {en} }