All 4 x 4 solutions of the quantum Yang-Baxter equation

Dear editor,

We would like to thank the reviewers for their careful reading of the manuscript and their comments. We tried to incorporate all the comments. In particular, referee 3 mentioned that we missed something. This urged us to recheck our computations. Indeed, we found some models in our Mathematica files that were not listed in the paper. We took the time to go through all are computations again and check against the manuscript to make sure it is complete now. To demonstrate this we added a new appendix where we spell out the details of all the different solutions that we get. Apart from the above, in accordance to the recommendations, we have made the following changes to the paper:

REFEREE 1: 1. Typo has been corrected. 2. Added requested citation in Sec.4. 3. Reformulated Prop.2.

REFEREE 2: 1. Reformulated the reference to the citation [6]. 1. Reformulated these sentences. 2. Added a bit more explanation in the beginning of Section 2. 3. Removed the confusing phrasing. 4. The definition of a constant solution can be found in the fourth paragraph of the introduc- tion. 5. There is no path from one solution to another one. Further; Equation (5) in the proof of Prop.1 is correct the way it is. Indeed, you can make the replacement as suggested, but it is a step easy enough to be done by the reader. 6. Added an explanation. 7. There is a hyperlink to the Appendix where the models are explicitly written down. 8. Yes, the highest order in question is always linear. 9. Added a comment in 2.1 to make more clear that higher order corrections can happen. 10. An overall table would not fit the page. But we included some flow charts in the appendix, to show starting from a constant model, which non-constant ones can be found. 11. Took out the confusing phrasing in the beginning of Sec 3. 12. The solutions are obtained by the method of section 2. From which constant model they stem, can be seen in the Appendix. 13. For non-regular models t is for example often zero, then the log would not be defined. 14. The concept of regularity has been introduced in the introduction. The way it comes up in case of the Lax operator is stated explicitly. As L is conceptually not that different from R, we see no reason to expand more on this. 15. The lower index notation has been introduced in the introduction, it is also very common in the literature. The argument you describe is written explicitly. 16. An explicit examples of this statement can be found in Sec. 5.2. 17. The citations are as found on the official websites. 18.-23. Typos corrected.

REFEREE 3: 1. Added a paragraph in the introduction to specify what we mean by classification 2. a) corresponds to our model RD. b) and c) were indirectly included in the result as we noted to have recovered all of the results of [2401.12710v2], but we should have listed them explicitly indeed. We have now included these models explicitly. 3. Our classification as noted in the introduction only considers one dimensional spectral parameters. We included your comment on higher dimensional spectral parameters, as another avenue to explore in the conclusion.

We hope this addressed all of your concerns. Thank you again for taking the time.

Kind Regards, Marius de Leeuw and Vera Posch

REFEREE 1: 1. Typo has been corrected. 2. Added requested citation in Sec.4. 3. Reformulated Prop.2.

REFEREE 2: 1. Reformulated the reference to the citation [6]. 1. Reformulated these sentences. 2. Added a bit more explanation in the beginning of Section 2. 3. Removed the confusing phrasing. 4. The definition of a constant solution can be found in the fourth paragraph of the introduc- tion. 5. There is no path from one solution to another one. Further; Equation (5) in the proof of Prop.1 is correct the way it is. Indeed, you can make the replacement as suggested, but it is a step easy enough to be done by the reader. 6. Added an explanation. 7. There is a hyperlink to the Appendix where the models are explicitly written down. 8. Yes, the highest order in question is always linear. 9. Added a comment in 2.1 to make more clear that higher order corrections can happen. 10. An overall table would not fit the page. But we included some flow charts in the appendix, to show starting from a constant model, which non-constant ones can be found. 11. Took out the confusing phrasing in the beginning of Sec 3. 12. The solutions are obtained by the method of section 2. From which constant model they stem, can be seen in the Appendix. 13. For non-regular models t is for example often zero, then the log would not be defined. 14. The concept of regularity has been introduced in the introduction. The way it comes up in case of the Lax operator is stated explicitly. As L is conceptually not that different from R, we see no reason to expand more on this. 15. The lower index notation has been introduced in the introduction, it is also very common in the literature. The argument you describe is written explicitly. 16. An explicit examples of this statement can be found in Sec. 5.2. 17. The citations are as found on the official websites. 18.-23. Typos corrected.

REFEREE 3: 1. Added a paragraph in the introduction to specify what we mean by classification 2. a) corresponds to our model RD. b) and c) were indirectly included in the result as we noted to have recovered all of the results of [2401.12710v2], but we should have listed them explicitly indeed. We have now included these models explicitly. 3. Our classification as noted in the introduction only considers one dimensional spectral parameters. We included your comment on higher dimensional spectral parameters, as another avenue to explore in the conclusion.