True and fake Lax pairs : how to distinguish them

The gauge-invariant description of zero-curvature representations of evolution equations is applied to the problem of how to distinguish the fake Lax pairs from the true Lax pairs. The main difference between the true Lax pairs and the fake ones is found in the structure of their cyclic bases. In this note, we use the gauge-invariant description of zero-curvature representations of evolution equations, which was developed in [1], to solve the problem of how to distinguish the fake Lax pairs, introduced in [2], from the true Lax pairs. This approach is algorithmic and exploits only the x-part of a studied Lax pair. The main difference between the fake Lax pairs and the true ones turns out to be clearly seen from the structure of their cyclic bases (we use the terminology of [1] and explain it step by step in what follows). Let us study a true Lax pair first. We take the true Lax pair of the modified KdV equation u t + u xxx − 6u 2 u x = 0 from [2]: