Solutions of the Yang-Baxter-like equation for a $3\times 3$ matrix of two Jordan blocks

Abstract

Let the Jordan canonical form of a $3 \times 3$ matrix $A$ be $J={\rm diag}(J_2[\lambda],J_1[\mu])$, where $\lambda\neq 0$ and $\mu\neq 0$. We find all the solutions of the Yang-Baxter-like matrix equation $AXA=XAX$ successfully by discussing whether $\lambda$ and $\mu$ are equal.