Another method for testing integrability is the Painleve test, see e.g. [these][1] lecture notes and references therein. It has some caveats: for example, certain changes of variables do not preserve the Painleve property. Yet another possibility for integrability testing is using higher symmetries, see e.g. [here][2]. 


  [1]: https://arxiv.org/pdf/solv-int/9804003.pdf
  [2]: https://mathoverflow.net/questions/302765/how-to-use-these-higher-symmetries-and-conservation-laws/309742#309742