Mathematical Institute,

Silesian University in Opava,

Na Rybnicku 1,

746 01 Opava,

CZECH REPUBLIC

E-mail: Peter.Sebestyen@math.slu.cz

**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.