On one ansatz for $\mathrm{sl}_2$-invariant $R$-matrices

The spectral decomposition of regular $\mathrm{sl}_2$-invariant $R$-matrices $R(\lambda)$ is studied by means of the method of reduction of the Yang–Baxter equation onto subspaces of a given spin. Restrictions on the possible structure of several highest coefficients in the spectral decomposition are derived. The origin and structure of the exceptional solution in the case of spin $s=3$ are explained. An analogous analysis is performed for constant $R$-matrices. In particular, it is shown that the permutation matrix $\mathbb P$ is a “rigid” solution.