Singularity analysis for integrable systems by their mirrors

We use the Lorenz system, the Rikitake model and the nonlinear Schrödinger equation to demonstrate that for completely integrable systems, there exist what we call regular mirror systems near movable singularities. The method for finding the mirror systems is very similar to the original Weiss et al's (1983 J. Math. Phys. 24 522-6) version of the Painlevé test. It tests the complete integrability and gives a systematic and conceptual proof that the formal Laurent series generated by the Painlevé test are convergent.