In addition to the answer by @issoloroap, the article *Prolongation structures of nonlinear evolution equations* by Allan Fordy ([here][1] is the Mathscinet link and [here][2] is its first page on Google books) explains why "good" zero-curvature representations should live in (semi)simple Lie algebras.

Another important point is that the spectral parameter should be essential, i.e., it should not be removable by gauge transformations, cf. e.g. [this][3] [paper][4] and references therein, and for the discussion of related issues in the case of dispersionless sysems in more than two independent variables see e.g. [this][5] [article][6], and [this][7] [one][8], and references therein. 


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=1090598
  [2]: https://books.google.com/books?id=eO_PAAAAIAAJ&pg=PA403
  [3]: https://arxiv.org/abs/0804.2031
  [4]: https://link.springer.com/article/10.1007/s10440-009-9450-4
  [5]: https://arxiv.org/abs/1401.2122
  [6]: https://link.springer.com/article/10.1007/s11005-017-1013-4
  [7]: https://arxiv.org/abs/1309.4993
  [8]: https://www.sciencedirect.com/science/article/pii/S0393044014001119