Solving the $4 \times 4$ quantum Yang-Baxter equation

In this paper, we complete the classification of 4×4 4 × 4 analytic solutions of the Yang–Baxter equation. Regular solutions were recently classified and in this paper we find the remaining non-regular solutions. We present several new solutions, then consider regular and non-regular Lax operators and study their relation to the quantum Yang–Baxter equation. We show that for regular solutions there is a correspondence, which is lost in the non-regular case. In particular, we find non-regular Lax operators whose R R -matrix from the fundamental commutation relations is regular but does not satisfy the Yang-Baxter equation. These R R -matrices satisfy a modified Yang–Baxter equation instead.