Аннотация:The equation XAX = AXA is called the Yang–Baxter-like matrix equation. Weexamine this equation for matrices of order 2, assuming that A is a nonsingular matrix; moreover,we are only interested in nonsingular solutions. Using a unified rule, one can associate each solutionwith a matrix that commutes with A, in other words, with an element of the centralizerMA of A. Noobvious reasons exist for two distinct solutions X1 and X2 to generate one and the same element ofMA. Nonetheless, all the solutions, and there are infinitely many of them, produce one and the samematrix in the centralizer. We give an explanation for this amazing fact.