Normal forms of sl(3)-valued zero curvature representation

Abstract:
One of the ways to overcome existing limitations of the famous Wahlquist-Estabrook procedure consists in employing normal forms of zero curvature representations (ZCR). While in case of $\mathfrak{sl}_2$ normal forms are long known, the next step is made in this paper. We find normal forms of $\mathfrak{sl}_3$-valued ZCR that are not reducible to a proper subalgebra of $\mathfrak{sl}_3$. We also prove a reducibility theorem, which says that if one of the matrix in a ZCR $(A,B)$ falls to a proper subalgebra of $\mathfrak{sl}_3$, then the second matrix either falls to the same subalgebra or the ZCR is in a sense trivial. In the end of this paper we show examples of ZCR and their normal forms.