Machine-Learning Search for Lax Connections

We apply a machine learning framework to search for Lax connections in two-dimensional non-linear sigma models using local current data. For the $SU(2)$ principal chiral model and the symmetric coset $S^2 = SU(2)/U(1)$, the method successfully recovers the full spectral-parameter families without using the known spectral curves as training targets. For the non-symmetric coset $T^{1,1}$, the optimization converges to reproducible low-loss maps that distill into a compact block-diagonal ansatz. However, analytic verification shows that this candidate is a ``fake Lax'' connection which satisfies on-shell flatness but fails to encode the two-dimensional equations of motion, whereas its point-particle reduction yields a genuine mechanical Lax pair. These results demonstrate that machine learning can effectively propose candidate ans\"atze and identify spectral structures, but low flatness loss alone does not certify genuine integrability, underscoring the necessity of analytic validation.