I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (3) and (5) in the paper: http://ac.els-cdn.com/S0375960199007665/1-s2.0-S0375960199007665-main.pdf?_tid=84c439b4-185f-11e6-9d95-00000aab0f27&acdnat=1463071101_131dcf6fe094a2dd3bd0a7458d1e12ac)
Obviously, the gauge transformation between them indicates that they generate the same soliton hierarchy, so my question is: why do these two spectral problems both exist if they are gauge equivalent. In other words, why have the two been studied separately? Is there any advantage of one over the other? Are there any known differences?
By the way I just started doing some research in this field and I would be grateful if anyone can give me their expert opinion on this issue.
Thank you.